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ISBN 978-5-87953-379-8 (Hapoolloe o6pa306Qllue)
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ISBN 978-5-87953-379-8 (Hapodl,oe 06Pa306Qllue)
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OT aBTOpOB .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
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 I. OCHoBHbIe nOHHTHSI reoMerpHH ........................... 8
H3MepeHHc OTpe3KOB ................................. 8
H3MepeHHe yrJIOB. CMe}l{HbIe H BepTHK3J1bHbIe yrJ1bl ..... 1 a
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3aJlallt c npaKTHlJeCKHM COep)l{aH"eM ................. 16
B3aHMHoe pacno.nO>KeHHe HeCKOJIbKHX npHMblX ......... 17
HliUl1lleHTHOCTb npHMblx H TOlJeK l-fa nJ10CKOCTH ......... 18
K 0 HT pOJl b H a 51 p a 6 OT a ................................. I 9
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IlpHXOJlHTCH npepblB3Tb TeMY «TpeyroJlbHHKH» H3JIO>KeHHe MaTepHa-
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a He B03BpamaTbC5I K 3TOH cpopMYJIe n031lHee. n pH 310M llOKa38TeJlb-
CTBO nepBoro IlpH3HaKa nO!106H5I TpeyrOJlbHHKOB MO)KHO OCYl1.leCTBHTb
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COCTaBJIelJHH 3K3aMel.laUIOHI1bIX 3an.aHHH 110 nJ13HHMeTpHH. TaKHM 06-
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llep>KalllaH 10 BapH3HToB, B Ka)K!lOM "3 KOTOpblX no 5 3a1l3HHH. Hc-
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HYIO) He n03BOJIHJIa CCPOpMHpOBaTb BapH3HT 113 5 3aJl3HHH. n03TOMY
KOHTpOJIbH3S1 K 2 06'beJlHHeHa C KOHTpOJlbHOH K 3, a KOHTpOJIbHaH K
4 06'beJlHHeH3 C KOHTpOJIbHOH K 5.
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5- 8 COCraBJIeHbl TaK, 1.JTo6bI YlJalllHeCH, OCBOHBI1JHe npoBepSieMbIM Ma-
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BbIweHHoro ypOBHSI CJ10)KHOCTH. 3TH TeCTbl npeJlHa3HaQeHbi MH Ylla-
LllHXCH, ycneBalOlllHx H8 «OTJlH1IHO». Ee30111H6oQHOe perneHHe Bcex
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no 3TOMY noco6HIO oy.nyr cnoco6cTBOBaTb nOBbIllleHHIO o6111ero ypOBHSI
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MCTpHH: B 2 qaCTHX», CM. [6] B cnHCKe JIHTeparypbl.
ABTOPbI 6JIaro.napHT peueH3eHTOB 33 npOIJTeHHe pYKonHCH Ii ueH-
IIIJle 3aMeQaHMSI.

8
9/. OCHOBHble nOHHTllH reOMerpHH
.
 1. OCHoBHble nOHSITHSI reOMeTpHH
1.1. H3MepeHHe OTpe3KOB
1. Ha oTpe3Ke AB, llJlHHa KOToporo paBHa 15, pacnOJIO>KeHbl TOIJKH
C Ii D r3K, liTO AC 6, BD 5. HaMJlHTe J1JIHHY oTpe3Ka CD.
2. Ha OTpe3Ke AB, J1JIHHa KOToporo paBHa 30, pacnOJlO>KeHbl T01.JKH
C H D T3K, liTO AC 14, BD 7. HaHJtHTe JlJlHHY oTpe3Ka CD.
3. Ha orpe3Ke AB, JlJIHHa KOToporo paBHa 144, pacnOJlO)KeHbl TOqKH
C Ii D TaK, lJTO AC 78, BD 87. HaHllHTe HJIHHY oTpe3Ka CD.
4. Ha oTpe3Ke AB, JlJIHH3 KOToporo paBHa 88, pacnOJlO>KeHbI TOqKH
C Ii D TaK, 4TO AC 55, BD 68. HaHJlHTe JtJIHHY oTpe3Ka CD..
5. Ha npHMoH B3HTbI TOlJKH A, B, C, TaKHe, liTO AB 34, BC 45.
4eMY MO>KeT 6bITb paBHa JlJIHHa oTpe3K3 AC?
6. Ha npSlMoH B3S1TbI TOlJKH A, B, C, TaKHe, 4TO AC 14, BC 20.
KaKoe HaHMeHbwee 3HalJeHHe MO>KeT HMeTb D.JIHI-I3 oTpe3Ka AB?
7. Ha npHMoM B3S1Tbl T04KH A, B, C, raKHe, 4TO AB 32, AC 18.
KaKoe H3H6oJlbWee 3HalJeHHe MO>KeT HMeTb LlJIHHa oTpe3Ka BC?
8. Ha OTpe3Ke MN, J1JIHH3 KOToporo paSHa 36, pacnOJIO)l{eHbl TOqKM
K H L T3K, liTO AIL 24, KL 9. 4eMY MO>KeT 6blTb paBHa J1JIHHa
oTpe3Ka KN?
9. Ha OTpe3Ke MN, JlJIHH3 KOToporo paBHa 49, pacnOJIO>KeHbl TOl.JKM
K H L T3K, liTO ML 35, KL 16. 4eMY MO}l{eT 6bITb pasHa J1JIHHa
OTpe3Ka KN?
1 O. Ha oTpe3Ke MN pacnOJIO)KeHbI TOllKH K H L TaK, liTO MK 18,
KL 15, LN 21. 4eMY MO}l{eT 6blTb pasHa MHH3 OTpe3Ka MN?
11. Ha oTpe3Ke MN pacnOJlO)KeHbl TOllKH K It L TaK, 1.JTO MK 22,
KL 17, LN 20. 4eMY MO>KeT 6bITb paSHa MHHa OTpe3Ka MN?

113MepeHHe Orpe3KOB
9
12. Ha orpe3Ke M N pacnOJlO>KeHbl TOIJKH K H L TaK, tfTO M K 32,
KL 34, LN 26. 4eMY MO)l{eT 6blTb pasHa D.JIHHa OTpe3Ka MN?
13. Ha OTpe3Ke MN pacnOnO>KeHbl TOlJKH K Ii L TaK, qTO MK 16,
KL 12, LN 10. 4eMY MO>KeT 6blTb pasHa D.JIHHa OTpe3Ka MN?
14. BHyrpM oTpe3Ka AB B3HTa TOlJK3 C TaK, LITO AC : AB 3 : 7.
HaHllHTe JUlHHY oTpe3Ka BC, eCJlH H3BeCTHo, liTO AB 56.
15. BHyrpH oTpe3K3 AB B3HTa TOlfKa C TaK, lJTO AC : AB 5 : 8.
HaHHHTe nJIHHY oTpe3Ka BC, eCJIH H3BeCTHO, lITO AC 40.
16. BHyrpH orpe3Ka AB B3HTa TOqK3 C TaK, tJTO AC : AB 3 : 5.
HaHllHTe .QJ1HHY oTpe3Ka AC, eCJlH H3BeCTHo, LITO BC 32.
17. BHyrpH oTpe3K3 AB B3HTa TOlJKa C T3K, liTO AC : BC 4 : 9.
HaHJtHTe J])IHHY oTpe3Ka AB, eC.flH H3BeCTHO, liTO AC 20.
18. BHYTPH OTpe3Ka AB B3HTa TOttKa C TaK, LITO AC : BC 2 : 7.
HattltHTe D.J1HHY oTpe3Ka AB, eCJlH H3BeCTHO, liTO BC 56.
19. BHyrpH oTpe3K3 AB B3HTa TOlJK3 C raK, liTO AC : BC 3 : 10.
HaHJ1HTe D.JIHHY oTpe3K3 AC, eCJIH H3BeCTHO, liTO AB 52.
20. BHYTPH oTpe3Ka AB B3HTa TOlJK3 C T3K, 4TO AC : BC 2 : 5.
HaMJtHTe JI.JIMHY oTpe3Ka BC, eCJlH H3BeCTHo, liTO AB 21.
21. Ha npSlMoH AB B3HTa TOllKa C TaK, liTO AC : AB 3 : 8. lIeMY
MO>KeT 6blTb paBH3 JlJIHH3 oTpe3K3 BC, eCJlH H3BeCTHO, QTO AC 12 ?
22. Ha npHMoH AB B3HTa TOQKa C TaK, liTO AC: AB 6 : 5. lIeMY
MO>KeT 6blTb pasHa JlJIHHa OTpe3K3 BC, eCJIH H3BeCTHo, QTO AC 18 ?
23. Ha npHMoH AB B3S1T3 TOqKa C TaK, lJTO AC : BC 5 : 3. 4eMY
MO>Ker 6bITb paBHa JlJIHH3 OTpe3K3 BC, eCJIH H3BeCTHo, liTO AB 16?
24. Ha npHMoM AB B3HTa TOllKa C TaK, LITO AC : BC 4 : 7. 4eMY
MO>KeT 6blTb paBAa JI)IMHa oTpe3K3 BC, eCJIH H3BeCTHO, liTO AB 33 ?
25. Ha nps:lMOH AB B3S1Ta TOllK3 C TaK, liTO AC : AB 3 : 5. 4eMY
M()>KeT 6bITb p3BHa JI)IHHa oTpe3Ka AB, eCJIH H3BeCTHO, QTO BC 20 ?

10

91. OCHOBHble nOHRTHR reOMeTpHH
26. Ha npSlMoH AB B3HTa TOqKa C TaK, tJTO AC : AB 7 : 5. 4eMY
MO>KeT 6blTb paBH8 llJIMHa oTpe3Ka AB, eCJIH H3BeCTHO, liTO BC 24 ?
27.* Ha OTpe3Ke AB pacnOJIO>KeHbl TOlJKH C Ii D TaK, 1.JTO TOlJKa D
Jle)f{HT BHyrpH oTpe3Ka BC. T01.JKH M H N - cepe,UHHbI OTpe3KOB AC
Ii BD. HaH,aHTe ltJIHHY oTpe3K3 MN, eCJIH AB 25, CD 11.
28.* Ha OTpe3Ke AB paCnOJIOiKeHbI TOlJKH C M D TaK, lITO TOlJK3 D
J1e)l{HT BHyrpH OTpe3Ka BC. TOlJKH M H N cepellHHbI OTpe3KOB AC
" BD. HaHllHTe }lJIHHY oTpe3K3 CD, eCJlH AB 36, MN 28.
29.* Ha OTpe3Ke AB pacnOJlO/KeHbl TOllKH C H D raK, liTO TOLJKa D
JIeiKMT BHYTPH OTpe3K3 AC. T04KH M H N cepe.nHHbl OTpe3KOB AC
" BD. HaH.LIJiTe JlJIHHY OTpe3K3 MN, eCJIH AB 57, CD 23.
30.* Ha OTpe3Ke AB pacnOJlOJKeHbl TOtJKH C H D T3K, liTO T04K3 D
JIe}l(HT BHyrpH OTpe3K3 AC. TOqKH M H N cepen.HHbI OTpe3KOB AC
H BD. HaH.QHTe JlJIHHY oTpe3K3 CD, eCJIH AB 60, MN 22.
1.2. 113MepeHHe yrJlOB. CMe>KHble H BepTHKaJlbHbie yrJlbl
31. B HyrpH pa3BepHyroro yr Jl3 AO B n pO BeJleH bl J1YIIH OC HOD,
TaKHe, liTO LAOC 73°, LBO D 85°. HaHJlHTe rpanycHYIO Mepy
yrJla COD.
32. BHyrpH pa3sepHyroro yr JIa AO B npOBeJteHbl J1YtlH DC HOD,
TaKHe, liTO LAoe 24°, LBO D 137 0 . HaHJtHTe rpallycHYIO Mepy
yrJla COD.
33. BHyrpH pa3BepHyroro yrJla AOB npOBelleHbl J1YLJH oe HOD,
TaKHe, liTO LAOC 106 0 , LBO D 17 4°. HaHllHTe rpaJ!ycHYIO Mepy
yrJIa COD.
34. BHYTPH pa3BepHyroro yrJIa AOB npOBeLteHbI JIYIJM DC 11 OD,
TaKHe, qTO LAOC 144 0 , LBOD 49°. HaHJlMTe rpaJlycHYIO Mepy
yrJla CODe

J13MepeHHe yr.nOB. CMeJKHble H BepTHKa.nbHble yr.nbl II
35. J1YIlI1 OA, DB, OC TaKOBbI, liTO LAOB 95°, a LBOe 70°.
4eMY MO)KeT 6blTb paBH3 rpaJlycHaS1 Mepa yrJIa AOe?
36. J1YlJH OA, OB, DC T3KOBbl, tITO LAOB 80°, a LBOe 150°.
4eMY MomeT 6blTb paBHa rpaJ!ycHa51 Mepa yrJl3 AOC, ecnH H3BeCTHo,
liTO LAOC < 180 0 ?
37. JIYlJH OA, DB, oe T3KOBbl, liTO LAOB 120°, a LBOe 90°.
4eMY MO)KeT 6blTb pasHa rpaJlYCHaSl Mepa yrJ18 AOC, eCJ1" H3BeCTHO,
4TO LAOC < 180 0 ?
38. BHyrpH pa3Bepl)'TOrO yr JIa AO B npOBeD-elbl J1YQH OC Ii 0 D,
TaKMe, LITO LAOC 100 0 , LCO D 30°. 4eMY MO>KeT 6blTb pasHa
r-p a n.YCHaS1 Mepa yr.na BOD?
39. BHYTPH pa3BepHYToro yrJI3 AOB npOBeneHbI JIYLJH oe HOD,
raKHe, LITO LAOe 120 0 , LCO D 80 0 . 4eMY MO>KeT 6blTb pasHa
rpallYCH3H Mepa yrJIa BOD?
40. BI-I}'TpH OAHoro 113 yr110B, 06pa30Bal-llblX JIYlJaMH OA HOB, npo-
selleHbI JIYlJH DC HOD, TaKHe, liTO LADC 50°, LCOD 40°,
LBDD 80°. 4eMY MO)KeT 6blTb paBll3 rpaltycH3H Mepa yrJIa AOB?
41. BHYTP" OJlHOrO 113 yrJlOB, 06pa30BaHHbIX JlytlaMH OA HOB, npo-
I\CJleHbl JIYllH OC HOD, T3KHe, liTO LAOC 90°, LeOD 45°,
LBOD 30°. 4eMY MO)KeT 6bITb paBH3 rpaJlycHaH Mepa yrJJ3 AOB?
42. JIyq OC JleJIHT pa3BepH}'TblH yrOJ1 AOB Ha JlBa yrJla, rpaJ!ycHble
MCpbl KOTOpblX OTHOCSlTC5I K3K 2 : 3. HaHllHTe rpallycHYIO Mepy 60JIb-
Illero H3 3THX yr J10B.
43. JIYlJ oe lleJIHT pa3BepH}'TblH yroJl AOB Ha llBa yrJl3, rpaJ1ycHble
t\1l\pbl KOTOpblX OTHOCHTCH KaK 4 : 5. HaH!lHTe rpal1ycHYIO Mepy MeHb-
Illl\rO H3 3THX yr JIOB.
.14. JIyq OC J1eJIHT pa3BepHYTblH yroJl AOB H3 ,QBa yrJl3,. a J1YlJH OL
II () J\;[ HBJlHIOTCH 6HcceKTpHcaMH YfJlOB AOe H BOC. 'uoK3}1{HTe, 4TO
 !\ I 0 L 90 0 .

12
.
 1. OCHoBHble nOHHTHH reOMeTpHH
45. BHYTPH pa3BepHYToro yrJ1a AOB npoBe,neHbl JlY4H OC H OD TaK,
liTO JIY4 OC JJe>KHT BHyrpH yrJla AOD. J1YlIH OL H OM HBnHIOTCSI
6HCCeKTpl1CaMH yrJIOB Aoe H BOD. HaHJlHTe rpallycHYIO Mepy yrJl3
1\10L, eCJII1 LCOD 78°.
46. BHYTPH pa3BepHyroro yrJ1a AOB npOBe,neHbl JlyqH OC H OD TaK,
4TO JIYLJ OC J1e)KHT BHYTPH yrJla AOD. J1Y l IH OL H ONI HBJlHIOTCSI
6HcceKTpHc3MH yrJIOB AOC H BOD. HaH.QHTe rpanycHYIO Mepy yrJl3
COD, eCJ1H LIYIOL 122 0 .
47.* BHYTPH pa3BepHyroro yrJIa AOB npOBeAeHbI JlYllH OC H OD
TaK, lJTO JIY l ) OD J1e>KHT BHyrpH yrJIa AOC. J1YlJH OL H OM HBnHIOTCSI
611cceKTpHcaMH yrJ10B AOC H BOD. HaHJlHTe rpanycHYIO Mepy yrJI3
1\10L, eCJIH LCOD 130 0 .
48.* BHyrpH pa3BepHYToro yrJla AOB npOBeeHbl J1Y4H OC H OD
TaK, liTO J1YI) OD JIe)KHT BHyrpH yrJIa AOC. JIYlJH OL H OM HBJlHIOTCH
6HcceKTpHc3MH yrJIOB AOC H BOD. HaHllHTe rpan.ycHYIO Mepy yrJla
COD, eCJlH LAfOL 15°.
49.* BHYTPH pa3BepHYToro yrJIa AOB npOBeJleHbI J1Y 1 IH OC H OD
TaK,4TO LCOD 40°, a yrOJI BOD B TPH pa3a 60J1bWe yrJ1a AOO.
4eMY MO>KeT 6blTb pasHa rpaJlYCHaH Mepa yrJIa AOC?
50.* BHYTPH pa3BepHYToro yr JIa AD B npOBel1eHbl J1Y4H OC HOD
TaK, 4TO LOOD 50°, a yroJ1 BOD B 4eTbIpe pa3a MeHbwe yrJIa AOO.
4eMY MO>KeT 6blTb paBHa rpan.ycHaH Mepa yrJIa BOD?
51.* BHYTPH pa3BepHYToro yrJl3 AOB nposeJ!eHbl JIY4H OC H OD
TaK, 4TO LOOD ' 60°, a rpaJlycHble Mepbl yrJIOB BOD H AOC OTHO-
CSITCH KaK 3 : 5 COOTBeTCTBeHHO. 4eMY MO)KeT 6bITb paBHa rpaJlycHaH
Mepa yr JIa AOC?
52.* BHyrpH pa3BepHYToro yrJl3 AOB npoBelleHbI JIY4H OC H OD
raK, liTO LCOD 75°, a rpaJlycHbIe Mepbl yrJloB BOD H AOC OTHO-
CHTCH KaK 4 : 11 COOTBeTCTBeHHO. 4eMY MO>KeT 6blTb pasHa rpa,nYCHaH
Mepa YfJIa BOD?

J13MepeHHe yrJ10B. CMe}/(Hble H BepTHK8J/bHbJe yrJlbJ
13
53. npOBeJIM OKPY}I{HOCTb C ueHTpOM B TOtlKe 0, a B HeM D.HaMeTpbI
AB H CD, TaKHe, 4TO LAOe + LBO D 306 0 . HaMJlHTe rpaJlycHYIO
Mepy yr JIB BOC.
54. npOBeJIH OKPY}I{HOCTb c ueHTpOM B T04Ke 0, a B HeM llHaMeTpbl
AB Ii CD, TaKHe, 410 LAOe + LBOD 3 · (LBOC + LAO D).
HaHllHTe rpaJlycHYIO Mepy yrJla BOC.
55. npOBeJ1H OKPY}I{HOCTb c ueHTpoM B TOtlKe 0, a B He" HH3Merpbl
AB,CD H EF, T3KMe, liTO LAOe 75°, LBOE 50°, a JIyq OE
ne>KHT BHYTP" yrJ1a BOC. Hal1J!HTe rpallycHYIO Mepy yrJla DOP.
56. npOBeJlH OKPY>KHOCTb c ueHTpOM B TOlJKe 0, a B HeM ,QH3MeTpbJ
AB, CD M EF, TaKHe, IlTO LAOe 60°, LBOE 100 0 , a nyq OE
Jle>KHT BHYTP" yrJla AOD. Hai1.nHTe rpaJ!ycHYIO Mepy yrJIa COF.
57. npOBeJ1H OKPY}I{HOCTb C ueHTpoM B TOllKe 0, a B HeH llH3Merpbl
AB, CD H EF, TaKHe, liTO LADe 110°, LBOE 120 0 , a JlY4 DE
JIe}l{l1T BHYTP" yrJIa AOC. HaHJlI1Te rpaJlycHYIO Mepy yrJ1a COF.
58. npOBeJlH OKPY}I{HOCTb C lleHTpOM B T04Ke 0, a B Hew !LHaMeTpbl
AB, CD H EF, T8KHe, liTO LADe 140°, LBOF 160 0 , a JlY4 OE
Jle)l{HT BHYTPH yr.na BOD. HaHl1HTe rpallycHYIO Mepy yrJI3 DOF.
59.* 113 TOlJKH 0 npOBelleHbl J1Y4H 0 A, 0 B, OC HOD, TaKHe, liTO
LAOB 90°, LBOe 160 0 , a LCOD 80°. 4eMY MO)l{eT 6blTb
pasHa rpaJ!YCHaH Mepa yrJIa AOD, eCJ1H LAOD < 180 0 ?
60.* 113 TOllKH 0 npOBelleHbl JIY4H OA, OB, DC " OD, T3KHe, liTO
LAOB 70°, LBOC 130 0 , a LCOD 100°. 4eMY MO}l{eT 6blTb
1)(lBHa rpanycHaH Mepa yr Jla AO D, eCJIH LAO D < 180 0 ?
1.3. PaccToSlHHe OT TOIIKH )1.0 npSlMoH.
PaccToSlHHe Me}l()Jy napaJIJleJlbHblMH npSlMbIMH.
fj 1. Ha nJ10CKOCTH OTMeqeHbl TOtlKH A, B H npHMaH l TaK, liTO npSlMaH l
Ill\pneHD.HKYJlHpHa OTpe3KY AB, HO He nepeceKaeT era, a paCCTOHHHe OT

14
.
J. OCHOBHble nOHRTJIH reOMeTpHH
TOLJKH A JlO npHMoH l paBHo 10. 4eMY MO>Ker 6blTb paSHO paCCTOHHHe
OT Cepelll1HbI OTpe3Ka AB O npHMOH l, eCJ111 M3BeCTHO, liTO AB 4 ?
62. I-Ia nJIOCKOCTH OTMe4eHbI TOLJKH A, B H npHMaH 1 raK, 1.JTO npHMaH I
nepneH)lJiKYJIHpHa orpe3KY AB, HO He nepeceKaeT ero, a paCCTOS1HHe 01
T04KH A AO npHMoH l pasHo 12. 4eMY MOiKeT 6blTb paBHo paCCTOHHHe
01' CepeltHHbl orpe3Ka AB JlO npfiMOH l, CCJJH H3BeCTI-IO, lITO AB 22?
63. Ha nJ10CKOCTH OTMeqeHbl TOlJKH A, B H npHM3H l raK, 4TO npHMaSi
1 nepneH.QHKYJIHpH3 OTpe3KY AB H nepeceKaeT era, a paCCTOHHHe OT
T04KH A AO npHMoH l paBIo 14. 4eMY paBHo paCCTOHI-IHe OT cepellHHbl
oTpe3Ka AB llO npSIMoH l, ecnH H3BeCTHO, 4TO AB 26?
64. Ha nJIOCKOCTH OTMellelbl TOllKH A, B 11 npHMaH l TaK, 4TO np5IM3S1
l nepneHLlHKYJ151pHa orpe3KY AB I nepeceKaer ero, a paccToHHHe or
TOtlKH A 110 npHMoH l pasHo 16. lJeMY paSHO paCCTOHHHe OT CepeJlHI1bl
OTpe3Ka AB 110 npSlMOH l, eCJlH H3BeCTHo, 4TO AB 42?
65. Ha nJIOCKOCTH OTMe4eHbi TOIIKH A, B H np51 Ma51 l T3K, liTO npHMaSi
l nepneHllHKYJ1HpHa OTpe3KY AB. 4eMY MO>Ker 6blTb paBlI3 JtJ1HI-I3 OT-
pe3Ka AB, eCJ1H paCCTOSlHHe OT TOllKH A J!O npHMoH l paBHo 18, a pac-
CTOHl-lHe OT CepeJlHHbI oTpe3Ka AB ,LtO npHMoH l pasHo II?
66. Ha nJ10CKOCTH OTMelJeHbI T04KH A, B H IlpHMaH l T3K, LJTO 11pSlM3S1
I nepneHJl.HKYJI5IpH3 oTpe3KY AB. 4eMY MO)l{er 6blTb paBHa JlJ1HH3 OT-
pe3Ka AB, eCJlH paccToHHHe OT T04KH A ,LtO npHMoH I paBHo 20, a pac-
CTOHI-IHe OT cepen.HHbl oTpe3Ka AB ,!to np5lMOH l pasHo 4?
67. Ha nJIOCKOCTH OTMeqeHbl TOIJKH A, B H np5lMaH l r3K, liTO fIpS1M351
1 nepneHJlHKYJlSIpHa OTpe3KY AB. 4eMY MO>KeT 6blTb paBHa )]JIHH3 OT-
pe3K3 AB, eCJIH paCCTOHHHe OT TOlJKH A no npHMoH l pasHo 22, a pac-
CTOHHHe or cepeJtHHbl oTpe3K3 AB no npHMoH 1 pasHo 28 ?
68. Ha nJIOCKOCTH .naHbI np5JMbIe l H p, napaJlJ1eJIbHbIe !lpyr JlPYry,
11 HeKOTopaH TOIIK3 A. 4eMY MO>KeT 6blTb paSHO paCCTOSlHMe Me>KJ!Y
npHMblMH 1 H p, eCJIH H3BeCTHO, 4TO paCCTOHHHe OT T01.JKH A 110 npHMoH
1 P8BHO 7, a paccToHHHe OT TOlJKH A JlO npS1MoH p pasHo 23?

PaCCTOHHHe Me>K.DJ' napaJ1.ne.nbHblMH npHMblMH
15
69. Ha nnOCKOCTH ,naHbl npHMble l H p, napaJIneJIbHble .llpyr JlPYry, Ii
HeKOTopaSi TOlJKa A. 4eMY MO>KeT OblTb paBHO paCCTOHHHe OT T01.fKH A
n.o npHMoH l, eCJIH H3BeCTHO, lJTO paccToSIHHe OT TOlJKH A ,llO npHMoM p
paBHo 25, a paccToHHHe Me>KJlY npHMblMH l H P paBHo 18?
70. Ha nJIOCKOCTM JlaHbl npHMble a, b, c, napaJIJleJIbHble J!pyr .aPyry.
OnpeJleJIHTe, KaKHM MO>KeT 6blTb paccToHHHe Me>KJIY npHMblMH a " c,
eCJlH H3BeCTHO, 4TO paccToHHHe Me>KUy npSlMblMM a H b paBHo 13, a
paccToHHHe Me>K){Y npSlMbIMH b H C pasHo 25.
71. Ha nnOCKOCTH .nBHbl npHMble a, b, c, napaJlneJIbHble Ilpyr llPYry.
OnpelleJIHTe, KaKHM MO>KeT 6blTb paccToHHHe Me}KJlY npSlMblMH a Ii C,
eCJlH H3BeCTHO, liTO paCCTOSlHHe Me>KUy npSlMbIMH a H b paBHo 30, a
paccToHHHe Me}l{Jty npSlMblMH b H C pasHo 14.
72. Ha nJIOCKOCTH .naHbl TOqKM A, B H npHMaH l. Onpe,QeJIHTe, K3Koe
H3HMeHbllJee 3HaqeHHe MO>KeT HMeTb llJIHH3 oTpe3K3 AB, eCJlH paccTo-
HHHe or TOlJKH A .no npHMoH l paBHo 14, a paccToHHHe OT TOqKH B JlO
npHMoH l paBHo 17.
73. Ha nJIOCKOCTH ):laHbl TOIJKH A, B H npSlM3S1 l. OnpeAeJIHTe, K3Koe
HaHMeHbwee 3Ha4eHHe MO>KeT HMeTb JlJIHHa oTpe3K3 AB, eCJIH paccTo-
fll-IHe OT T01.JKH A )!O npHMoH l paBHo 70, a paccToHHHe OT TOlJKH B .no
IlpSlMOH 1 pasHo 33.
74.* Ha nJIOCKOCTH }laHbl qeTbIpe pa3JIHlJHbie npHMbIe a, b, C H d,
IlapaJIJ1eJIbHble llpyr D-pyry. OnpelleJIHTe, KaKHM MO>KeT 6blTb paccTo-
SIII11e Me>KllY npHMblMH a Ii d, eCJIH H3BeCTHO, liTO paCCTOflHHe Me}l{J1y
111)flMbIMH a H b paBHo 8, paccToflHHe Me>KllY npSlMbIMH b Ii C paBHo 12,
;1 paccToSlHHe Me>KAY npSlMblMH C H d pasHo 16.
75.* Ha nJIOCKOCTH n.aHbl qeTbIpe pa3JIHIIHbIe npSlMble a, b, C H d,
11:IIJ3JIJleJIbHble Llpyr ,apyry. OnpeD.eJIHTe, K8KHM MO)KeT OblTb paccTo-
llllle Me>KJJY npHMblMH a H d, eCJlIf H3BeCTHO, liTO paCCTOSIHHe Me>K.U}'
111)IMbIMH a Ii b pasHo 3, paccToSlHHe Me>KJlY npHMbIMM b 11 C paBHo 24,
I 1)lCCT0S1HHe Me}l{J!Y npHMblMH C Ii d paBHo 27.

16
..
 J. OCHOBHbJe nOHHTHH reOMeTpHH
1.4. 3aAaliH c npaKTHlieCKHM COAep>KaHHeM
76. B HeKOTopOM nocfuIKe BD-OJIb HOBOH YJIHl.(bI Heo6xollHMO YCTaHO-
BHTb onopbI nHHHH 3JIeKTpOnepellaIJH (onopbl Jl30). 00 KJIHMaTHqe-
u v ..
CKHM YCJIOBHHM MeCTHOCTH, B KOTOpOH pacnOJIO>KeH llaHHbIH nOCeJlOK,
paccToHHHe Me>KJIy onopaMH Jl30, a TalOKe paCCTOSIHHe 01 KOHeBbIX
T04eK JI 3 n 40 6JIH}I{aHWeH onopbl, ,aOJI)I{HO He npeBbIllJaTb 25 MeTpoB.
KaKoe HaMMeHbwee lJHCJIO onop nOTpe6yeTcH YCTaHoBHTb B,llOJlb 3TOH
YJIHUbl, eCJlH ee J]JIHH3 cocr3BnHeT 800 MeTpoB?
77. OTBeTbTe Ha Bonpoc npeJlblJ1YLLleH 3allalfH, eCJlH llJ1HHa YJIHUbl co-
CTaBJIHeT 1250 MerpoB, a paccToSlHHe Me>KJlY onopaMH J13 n 11 paccTo-
HHHe OT KOHueBblX TOlJeK J13 0 JlO 6JIH)KaHWeH onopbl LlOJl>KHO He npe-
BbIlll3Tb 30 MeTpoB.
78. nYHKT A coe,nHHHJlH c nYHKTOM B LloporoH, npOTH)I{eHHOCTb KOTO-
pOM COCTaBHna 5 KM 300 MeTpoB. Heo6xoJlHMO npoBecTH 03eJ1eHeHHe
U U ..
3TOH lloporH, nOC311HB B)lOJ1b Ka>K!lOH ee CTOpOHbI no TPH pHJla ca*eH-
ueB ltepeBbeB T8K, QTo6bl B K3>KllOM pHD.Y paccToHHHe Me}I{Jl.Y COCe]lHH-
MH C3>KeHuaMH He npeBblWaJIO 6 MeTpoB H paccToHHHe OT KOHUOB 110- .
porH AO 6JIH)I{3HWerO Ca>KelLla TaK>Ke COCTaBJISlJlO He 60JIee 6 MeTpoB.
KaKoe HaHMeHblllee 1.JHCJlO C3)KeHueB Tpe6yeTcSI J1JI5I 3Toro nOC3DJiTb?
(npH BblllHCJIeHHH pa3MepaMH caMHX Ca}l{eHueB npeHe6pelJb).
79. OTBeTbTe H3 Bonpoc npellbID-Yl.lleH 3a.aallH, eCJIH npOTSI)I{eHHOCTb
lloporH COCTaBJIHeT 4 KM 900 MeTpoB, B,llOJIb Ka>KJ{oH ee CTOpOHbI HY>K-
HO nocaLlHTb no .D.Ba pHJ{a C3>KeHLleB ,nepeBbeB, a paccToHHHe Me)l{llY CQ-
cellHHMH Ca}l{eHuaMH Ii paccToHHHe OT KOHLlOB JlOpOrH JlO 6JIH}I{aHlllerO
Ca}KeHua llOJI>KHO He npeBbIwaTb 8 MeTpOB.
80. Tpe6yeTcH nOCTpOHTb MOCT Qepe3 peKY WHPHHOH 165 MeTpoB TaK,
lJTo6bl KOHLlbI MOCTa 6bIJIH Y,QaJleHbl OT 6eperoB peKH He MeHee, qeM H3
30 MeTpoB. KaKoe HaHMeHblllee lJHCJIO onop ,nOJI}I{eH HMeTb TaKOH MOCT,
eCJlH paccToHHHe Me}l{n.y COCe,QHHMH onopaMH MOCTa H paccToSlHHe OT
K3}1{JlOrO 113 KOHUOB MOCT3 o 6JIH>KaHWeH onopbl .nOJ]}I{HO 6blTb He 60-
nee 20 MeTpoB, a wpHHa K3}1{llOH onopbl COCT8BJlSleT 1,5 MeTpa?

B3aHMHoe paCnOJlO)/(eHHe HeCKO.RbKHX npHMblX
17
81. B MHOr03Ta)KHOM >KHJIOM JlOMe BbICOTa UOKOJISI (paccToSlHHe OT
3eMJlH !to nOJIa nepBoro 3T3)1{a) paSHa 1,2 MeTpa, paccToSlHHe OT nOJIa
JlO nOTOJlKa H3 Ka)KllOM 3Ta>Ke paBHo 2,8 MeTpa, a BbICOT3 nJIHT nepe-
KpblTHSI Me>KnY 3T3>KaMH pasHa 22 CM. BbllJHCJlHTe paCCTOSlHHe OT 3eM-
nH JIO nOJI3 7 -ro 3T3JKa. OTBeT BbIpa3HTe B MeTpax.
1.5. B3aHMHoe pacnOJlO}l(eHHe HeCKOJlbKHX npHMblX
82. Ha K3Koe tlHCJIO 4acreH Moryr pa3lleJIHTb nJIOCKOCTb TPH npHMble?
83. Ha KaKoe H3HMeHbllJee lIMCJlO qaCTeH MO>KeT 6blTb pa3.QeJleHa nJloc-
KOCTb qeTbIpbMSI npHMbIMH?
84. Ha KaKoe HaH60Jlblllee tJHCJlO liaCTeH MO)KeT 6blTb pa3,UeJIeHa nJIOC-
KOCTb lIeTblpbMSI npSlMbIMH?
85.* Onpen.eJIHTe, H3 KaKoe HaH60JIbWee LJHCJlO qaCTeM MOryT pa36HTb
n.nOCKOCTb: a) ceMb npHMblx; 6) n npHMblx ( n - HaTypaJlbHoe qHCJIO).
86.* HeCKOJIbKO npHMblX npOBeJIl1 T3K, LITO OHH pa3D-eJIHnH nJlOCKOCTb
H3 10 lJaCTeH. CKOJlbKO npSIMblx MOrJIO 6blTb npoBel1eHo?
87.* HeCKOJIbKO npHMbIx npOBeJIH T3K, I.JTO OHH pa3JleJIHJIH nJlOCKOCTb
I-fa 12 qaCTeH. CKOJIbKO npHMblx MOrJlO 6blTb npOBeJleHo?
88.* HeCKOJlbKO npHMbIX npOBeJIH TaK, liTO OHM pa311eJIHJIH nnOCKOCTb
Iia 90 qaCTeM. KaKoe H311MeHblllee LJHCJIO npSlMbIx Morno 6blTb npoBe-
JLCHO?
1.6. HHlI.HAeHTHocTb npSlMblX H TOlieK
(I1HUH,aeHTHocTb M3TeMaTHQeCKHH repMHH, ynoTpe6nHeMblH lIJISI
()003HaQeHMSI OTHoweHHH npHHallJIeJKHOCTH Me>K1lY OCHOBHblMH 06'beK-
'I (IMH reOMeTpHH TOtIKaMH, npHMbIMH, nJIOCKOcTSlMH.)
N9. KaKoe H3M6oJlblllee qHCJlO TOlJeK nepeCeqeHHH MO)KeT 6blTb y qe-
11»II)ex pa3JIHQHbiX npHMbIx?

18
J. OCHOBHble nOHHTHfi reOMerpHH
.
90. KaKoe HaH60JlbIllee lJHCJIO TOqeK nepecelleHHH MO>KeT 6bITb Y ,lle-
CSlTM pa3JIHtlHbIX npHMbIx?
91.* Ha nJlOCKOCTH OTMe1.JeHO 7 TOlJeK H npoBeAeHo HeCKOJIbKO npH-
MbIX T3K, tJTO Qepe3 Ka>KJlYIO "3 OTMeqeHHblX T04eK npOXOllHT pOBHO llBe
H3 3TliX npSlMblX. KaKOe HaHMeHblllee LJHCJlO npSlMbIX MOrJIO 6bITb npo-
Be.neHO?
92.* OTBeTbTe H8 Bonpoc npeHbl,QYllleH 3a,l1atJH, eCJIH KOJIHQeCTBO OT-
MeqeHHblX TOLJeK paBHo 37.
93. Ha nJlOCKOCTH 3a.llaHbI 6 T04eK o6111ero nOJlO>KeHHH (T.e HHKaKHe
TPH 113 HHX He Jle>KaT H3 OJlHOH npHMOH) H LJepe3 K3}KJtble llBe 113 3THX
TOlJeK npOBe.n.eHa npSlMafi. CKOJlbKO npHMbIX npOBeeHO?
94. a) CKOJlbKO D.HarOHaneH HMeeT npOH3BOJlbHbiM rnecTHyroJlbHHK?
6) CKOJ1bKO llHarOHaJleH HMeeT npOH3BOJIbHbIH n -yroJlbHHK?
95. Cyr.uecTByeT J1H MHoroyrOJIbHHK, 1.JHCJIO JlHarOHaJIeH KOToporo paB-
HO 12?
96. YKa>KHTe Bee 3HaqeHHH n, DJISI K3>KQOrO H3 KOTOpblX QI1CJlO .HHarO-
H3J1eH n - yro.nbHHKa paBHO 2n.
97.* Ha nJIOCKOCTH npOBeJIH 5 npHMblx o6111ero nOJIO>KeHHH (HHKaKHe
l{Be H3 HHX He napaJIJIeJlbHbI M HHK3KHe TPH 113 HHX He npOXOJ{HT Ilepe3
OJlHY H TY me TOqKY). 3aTeM H3 3TOH nJIOCKOCTH OTMeTHJIH HeCKOJIbKO
u ..
TOlJeK T3K, liTO H3 Ka>K,UOH H3 npOBe.aeHHbiX npS1MblX OKa3aJ10Cb POB-
HO no TP" OTMelJeHHblX TOqKH. KaKoe HaHMeHbwee qHCJIO TOqeK Mor JIO
6blTb OTMeqeHo?
98.* OTBeTbTe Ha Bonpoc npellbIJl)'llleH 3a!lalJH, eCJlH KOJIHLJeCTBO npo-
BeJleHHblx npHMbIX paBHo 6.

KOHTpOJlbHan pa60Ta
19
KOHTpOJlbHaH pa60Ta
PeKOMeHJlyeMoe BpeMH JlJ1H BblnOJlHeHHH pa60Tbl 40 MHHyr.
PeKoMeHnyMafi WKaJIa OlleHHBaHHH pe3YJlbTaToB:
.... 0-1 2 3 4-5
Lllllcno BepH blX peweH.,,1/1
WKonbHa ou.eHKa 2 3 4 5
BapHaHT 1
1. Ha oTpe3Ke AB, JlJIHH3 KOToporo pasHa 39, pacnOJIO)Kelibl T04KH C
I D T3K, liTO AC 15, BD 13. HaCtllHTe lL1IHHY oTpe3K3 CD.
2. BHyrpl-f OTpe3Ka AB B351Ta TOlJKa C T3K, tlTO AC : AB 14 : 15.
I-laHJlHTe J])1HHY oTpe3Ka BC, eCJIH H3BeCTHo, liTO AC 70.
3. .nytlH OA, OB, OC T3KOBbl, liTO LAOB 111°, a LBOC 22°.
LleMY MO)KeT 6blTb paBHa rpallycI1aH Mepa yrJla AOC?
4. npOBeJlH OKPY)KHOCTb c ueHTpoM B T04Ke 0, a B HeM 1!HaMeTpbl AB
If CD, TaKHe, liTO LAOC + LBO D 296 0 . HaHHHTe rpaJlycHYIO Mepy
yrJ13 BOC.
5. B noceTIKe BllOJIb HOBOH YJIHUbI Heo6xo!lHMO ycraHoBHTb onopbl JIH-
IIIIH 3JIeKTpOnepell3 l lH (onopbl JI3n). no KJlHMaTHQeCKI1M YCJIOBHHM
U U ..
MlCTHOCTH, B KOTOP0i'. pacn0J10>KeH lt3HHbIH nOCeJlOK, paccToHHHe Me)K-
JIY onopaMH .n3n, a TalOKe paCCTOSIHHe 01 KOHueBblX TOLJeK .n3n ,no
c )JII'J)KaHWeH onopbI, ,UOJI/KHO He npeBblwaTb 20 MeTpoB. KaKoe HaM-
Ml'llbwee lfHCJIO onop nOTpe6yeTcS1 YCT3HOBMTb B,llOJlb 3TOH YJlHUbI, ec-
.1111 ce JI.JIHHa COCTaBJISleT 850 MeTpoB?
BapHaHT 2
I. Iia OTpe3Ke AB, J1JIHH3 KOToporo pasHa 18, pacnOJIO>KeHbl TOtJKI1 C
II I) TaK, LITO AC 9,5, BD 7 ,5. H3HAHTe JlJIHHY oTpe3K3 CD.

20
 J. OCHOBHbJe nOHHTHH reOMeTpHH
-
2. BHYTPH oTpe3Ka AB B3HT3 T04Ka C TaK, liTO AC : AB 3 : 5.
HaHllHTe DJlHHY oTpe3Ka AB, eCJIH 3BeCTHO, liTO BC 48.
3. J1Y4H OA, DB, OC TaKOBbl, liTO LA DB 49°, a LBOC 125°.
4eMY MO>Ker 6blTb pasHa rpaJlYCHafi Mepa yr JJ3 AOC?
4. npOBeJIH OKPYiKHOCTb c ueHTpoM B T04Ke 0, a B HeH llHaMeTpbl AB
H CD, T3KMe, QTO LAOC+LBOD 234°. HaHJlHTe rpall)'cHYIO Mepy
yr Jla BOC.
5. B nOCMKe BJlOJlb HOBOH YJlHUbI Heo6xollHMO YCTaHOBHTb onopbl J1)1-
HHH 3J1eKTpOnepe1l3 t lH (onopbl .n3fl). flo K.nHMaTHlleCKHM YC.nOBHHM
U U ..
MeCTHOCTH, B KOTOpOH pacn0J10>KeH Jl3HHbIH nOCeJIOK, paccToflHHe MelK-
llY onopaMH JI3n, a TaK)Ke paccToHHHe OT KOHueBblX TOlJeK J13fll1o
6JIHiKaHWeii onopbl, )lOJIiKHO He npeBbIwaTb 30 MeTpoB. KaKoe H3H-
MeHbwee 4HCJ10 onop norpe6yercH YCTaHOBI1Tb Bllonb 3TOH YJlHLlbl, ec-
JIH ee llJ1HH3 COCT3BJlHeT 1 KM?
BapHaHT 3
1. Ha orpe3Ke AB, J]JIHHa KOToporo paBl-la 167, pacnOJIO)KeHbl T04KM
C Ii D TBK, LITO AC 97, BD 79. HaHJ!HTe HJIHHY oTpe3Ka CD.
2. BHyrpH oTpe3Ka AB B3f1Ta T04Ka C raK. 4TO AC : BC 20 : 13.
HaHJ!HTe J])1HHY orpe3Ka BC, eCJIH H3BeCTHO, liTO AB 132.
3. BHyrpH pa3BepHyroro yrJla AOB npOBeJleHbl JIYllH DC HOD, ra-
KHe, liTO LAOC 112 0 , LCOD 21 0 . 4eMY MO>KeT 6blTb pasHa
rpallYCHaH Mepa yrJIa BOD?
4. flpOBeJ1H OKPY>KHOCTb C LleHTpoM B TOqKe 0, a B He" D.HaMeTpbl AB
H CD, TBKHe, 4TO LAOC+LBOD 5.(LBOC+LAOD). HaHLlHTe
rpaJlycHYIO Mepy yrJIa BOC.
5. flYHKT A COeD.I1HHJlH c nYHKTOM B lloporoH, npOTH)I(eHHOCTb KOTOPOH
COCTaBHJIa 5 KM 500 MeTpoB. Heo6xoD.HMO npoBecTH 03eJ1eHeHHe 3TOH

KOHTpOJlbHan pa60r8
21
lloporH, nOCallHB Bl{OJlb Ka>K,QOH ee CTOpOHbl no JlBa pHlla calKeHLleB Jle-
peBbeB TaK, IJTo6bl B Ka}KllOM pSlJ!Y paCCTOSlHHe MelKJlY COCe)1HHMH ca-
)l{eHuaMH He npeBblWaJIO 6 MerpoB H paccToS1HHe OT KOHUOB !lOpOrH lI.O
6JIHIKaHWero calKeHua TaK>Ke COCT3BJlHJlO He 6o.nee 6 MeTpoB. KaKoe
H3HMeHbwee QHCJIO Ca)f{eHueB rpe6yeTcH MH 3Toro nOCaJlHTb? (npH
Bbl t IMCJ1eHHH pa3MepaMH C3MI1X calKeHLleB npeHe6pe1Jb).
BapHaHT 4
1. Ha orpe3Ke AB, ,UJlMHa KOToporo pasHa 31, paCnOJlOlKeHbI TOlJKH C
H D TaK, liTO AC 20, BD 15. HaHJlHTe J1.JIHHY oTpe3Ka CD.
2. BHYTPH oTpe3K3 AB B3S1Ta TOlJKa C TaK, liTO AC : BC 3 : 7.
l-IaHD.I1Te lL1lHHY oTpe3Ka AC, eCJIH H3BeCTHO. liTO AB 60.
3. BHYTpH pa3BepHYToro yrJIa AOB npoBe.neHbl JIYI.JH OG HOD, Ta-
I{He, lITO LAOe 105 0 , LCD D 85°. 4eMY MOlKeT 6blTb paBH3
rpaJI.ycHaSl Mepa yr JIa BO D ?
4. npOBeJ1H OKPYIKHOCTb C LleHTpoM B TOqKe 0, a B HeH llHaMeTpbl AB
II CD, T3KHe, liTO LAOC+LBOD 7.(LBOC+LAOD). HaHJlHTe
rrJanycHYIO Mepy yrJIa BOG.
5. nYHKT A COeHHHJIH c nYHKTOM B JlOpOrOH, npOTH>KeHHOCTb KOTOpOH
l.()CTaSHJI3 6 KM 700 MeTpoB. Heo6xoJIJ;fMO npoBecTH 03eJ1eHeHHe 3TOH
U ..
Jlf)pOrH, noca,nHB BJlOJlb Ka)KJlOH ee CTOpOHbI no TPH pHlla caiKeHueB lle-
I )CBbeB TaK, lJTo6bl B Ka>K-QOM pSln.y paccToHHHe MelKJ1Y COCeJlHHMH ca-
II\CllllaMH He npeSblWaJIO 8 MeTpoB H paccToHHHe OT KOHLlOB lI.OpOrH ]!O
fi.lllllKaHWerO Ca)KeHua TaK)Ke COCTaB.nHJlO He 60JIee 8 MerpoB. KaKoe
11.lliMeHbwee 4HCJIO Ca/KeHueB Tpe6yeTcH J1.JIH 3Toro nocallHTb? (npH
l'I.IIIHCJIeHHI1 pa3MepaMH caMHX Ca)KeHueB npeHe6pe4b).
BapHaHT 5
I. II a npHMoH B3stTbl T04KH A, B, C, TaKHe, liTO AC 15, BC 13.
1\ flf<oe H3HMeHblllee 3HaqeHHe MO>KeT HMeTb D/lHHa oTpe3Ka AB?

22
"
 J. OCHOBHbJe nOHHTHH reOMerpHH
2. Ha np5lMOH AB B3HTa T04K3 C TaK, liTO AC : AB 4 : 15. 4eMY
MO)Ker 6blTb paBH3 J.tJIHH3 oTpe3Ka BC, eCJlH H3BeCTHO, QTO AC 20?
3. BHYTPI1 yrJIa, o6pa30B3HHoro JIY43MH OA HOB, IlpOBeJleHbI JIYllH
OC HOD, TaKHe, liTO LAOC 35°, LCOD 15°, LBOD 55°.
4eMY MO}KeT 6bITb paBHa rpaJlycHaH Mepa yrJIa AOB?
4. npOBeJlH OKPY>KHOCTb C lleHTpoM B TOqKe 0, a B HeH D.HaMeTpbI AB,
CD H EF, TaKMe, liTO LAOC 60°, LBGE 70°, 3J1YQ OE JIe>KHT
BHYTPH yrJIa BOC. HaHlll1Te rpa.nycHYIO Mepy yr.na DOF.
5. B MHor03TaIKHOM >KH.flOM }!OMe BblCOTa LlOKOJIH (paccToSIHHe OT 3eM-
JlH 110 nOJIa nepBoro 3Ta>Ka) paSHa 1,4 MeTpa, paCCTOSII-IHe OT nOJIa !to
nOT0J1K3 Ha K3>KllOM 3Ta>Ke pasHo 2, 7 MeTpa, a BblCOTa nJIHT nepeKpbl-
TH5I Me)K!1Y 3Ta>KaMH paBHa 21 CM. BbI4HCJIHTe paCCTOHHHe 01 3eMJlH AO
nOJl3 5 -fa 3T3>Ka. OTBeT Bblpa3HTe B MeTpax.
BapHaHT 6
I. I-Ia npHMoH B3HTbI TOllKH A, B, C, T3KHe, liTO AB 31, AC 13.
KaKoe H3116oJ1bWee 3Ha4eHHe MO>KeT HMeTb )lJlHH8 oTpe3K3 BC?
2. Ha npHMoH AB 83f1T3 TOllKa C T3K, 4TO ..4C : AB 9 : 7. 4eMY
MOlKeT 6blTb pasHa lIJIHHa OTpe3Ka BC, eCJIH H3BeCTHO, 4TO AC 36?
3. BHyrpH yrJIa, o6pa30BaHHoro J1Y43MH OA HOB, npOBeJ1eHbI JlytJH
OC HOD, TaKHe, liTO LAOC 65°, LCOD 40°, LBOD 35°.
4eMY MO}l{eT 6bITb pasHa rpallYCHaH Mepa yr.na AOB?
4. np()BeJIH OKPY>KHOCTb C LleHTpoM B T04Ke 0, a B HeM JlHaMeTpbI AB,
CD H EF, TaKHe, liTO LAOC 32°, LBOE 123 0 , a JIyq OE JIe}KHT
BliYTPH yrJIa AOD. HaHllHTe rpaJlycHYIO Mepy yrJl3 COF.
5. B MHOr03Ta>KHOM >KHJlOM !lOMe BbICOTa UOKOJIH (paCCTOHHHe OT 3eM-
J1H AO n0J13 nepBoro 3T3>Ka) paBH3 1,3 MeTpa, paccToSlHHe OT nOJIa 110
nOTOJIKa Ha Ka)KLlOM 3T3>Ke pasHo 2, 75 MeTpa, a BbICOTa nJIMT nepeKpbl-

KOHTpOJlbHaH pa60r8
23
THSI Me>KJlY 3Ta)l{3MH paBHa 23 CM. BbIlJHCJlHTe paCCTOHHHe OT 3eMJIH .no
nOJI8 6-ro 3T3)1{3. OTBeT Bblpa3HTe B MeTpax.
BapHaHT 7
I. Ha orpe3Ke MN, JI.JIHH3 KOToporo paSHa 45, pacnOJlO>KeHbI TOqKH
K H L r3K, liTO ML 20, KL 15. 4eMY MO>KeT 6blTb paBH3 JI.JI"H3
oTpe3K3 KN?
2. Ha npHMoH AB B3HT3 TOtJK3 C T8K, liTO AC: AB 7 : 3. 4eMY
MO)KeT 6blTb paBHa JI.JIHHa oTpe3K3 AB, eCJIH H3BeCTHO, 1.JTO BC 40 ?
3. BHyrpH pa3BepHyroro yrJIa AOB npoBelleHbI nY1JM OC Ii OD TaK,
liTO JIY4 OC ne)l{HT BHyrP" yrJl3 AOD. JIY4H OL H OM SlBJlS1IOTCH
6HcceKTpHcaMH yrJloB AOC 11 BOD. HaH,QHTe rpan.ycHYIO Mepy yrna
/\10L, eCJ1H LCOD 102°.
4. BHYTPH pa3BepHYToro yrJIa AOB npoBelleHbI nyl.JH OC H OD TaK,
Il'ro LOOD 99°, a yroJl BOD B llBa pa33 MeHbwe yrJl3 AOC. 4eMY
r\1()/KeT 6bITb paBHa rpaJIYcHaH Mepa yrJIa BOD?
fi. HeCKOJ1bKO npHMblx npOBeJlH TaK, qTO OHM pa311eJIHJlH nJIOCKOCTb H3
'f tlaCTeH. CKOJIbKO npHMbIX MOrJIO ObITb npOBeJ1eHO?
BapHaHT 8
I. I-Ia OTpe3Ke MN, DJ]HH3 KOToporo paBH3 24, pacnOJIO)l{eHbl TOIJKH
/\' II L raK, qTO M L 15, KL 11. 4eMY MO)l{eT 6blTb pasHa .lL1IHHa
It II)C3Ka KN?
:'. II€J npHMOH AB B3S1T3 TOlJKa C TaK, liTO AC : BC 2 : 7. 4eMY
'( lil\cr 6blTb pasHa J]JIHH3 OTpe3Ka BC, eCJlH H3BeCTHO, QTO AB 45 ?
:1. I\IIYTPH pa3BepHyroro yrJI3 AOB npoBe.neHbl J1ytJH DC Ii OD TaK,
III () JIYll OC JIe)l{HT BHyrpH yr Jl3 AO D. J1 YQH 0 L H OM HBnHIOTCSI
f .11.('('KTpHCaMH yrJIOB AOC H BOD. HaHD-HTe rpan.ycHYIO Mepy yrJla
f " ) I J, eCJIH LMOL 98 0 .

24
..
 J. OCHOBHble nOHHTHH reOMeTpHH
4. BHyrpH pa3BepHyroro yrJla AOB npOBel!eHbl JIyqH OC H aD TaK,
liTO LCOD 30°, a yron BOD B qeTblpe pa3a 60JIbWe yrna AOC.
4eMY MO>KeT 6blTb paBHa rpaJlYCHaSl Mepa yrJIa AOC?
5. HeCKOJIbKO npSlMblX npOBeJIH TaK, 1.JTO OHM pa3,ne.nHJlH nJIOCKOCTb Ha
8 qaCTeH. CKOJ1bKO npSlMbIx MOrJIO 6blTb npOBeJleHo?
BapHaHT 9*
1. Ha oTpe3Ke M N pacnOJIO>KeHbI TOIJKH K Ii L TaK, liTO M K 44,
KL 33, LN 22. 4eMY MO>KeT 6blTb pasHa JlJIHHa OTpe3Ka MN?
2. Ha OTpe3Ke AB pacnOJIO>KeHbI TOl.lKH C H D TaK, 4TO TOlfK3 D ne-
)l{HT BHyrpl1 OTpe3K3 BC. TOqKH M 11 N cepellHHbI OTpe3KOB AC H
BD. HaHJlHTe lVIHHY oTpe3Ka CD, eCJlH AB 54, MN 32.
3. BHyrpH pa3BepHyroro yr JIa AO B npOBel{eHbl JIyqH OC HOD TaK,
liTO nYIJ OC JIe>KHT BHyrpH yrJIa BOD. JIYlIH OL H OM HBJIHIOTCSI
6HcceKTpHcaMH yrJloB AOC Ii BOD. HaHJlMTe rpa.nycHYIO Mepy yrJIa
MOL, eCJIH LCOD 100°.
4. BHyrpH pa3BepHyroro yrna AOE npOBeJleHbI JlY1JH OC H OD TaK,
tJTO LCOD 70°, a rpaJtycHble Mepbl yrJIOB BOD H AOC OTHOCHTCS1
KaK 2 : 3 COOTBeTCTBeHHO. 4eMY MO>KeT 6blTb pasHa rpa,nYCH3H Mepa
yrJI3 AOC?
5. HeCKOJlbKO npHMbIX npOBeJIH T3K, 1JTO OHH pa3HeJIHJlH nJ10CKOCTb H3
990 qaCTeH. KaKoe H3HMeHbwee lJHCJIO npHMbIx MOrJIO 6blTb npoBee-
HO?
BapHaHT 1 0*
t. Ha OTpe3Ke M N pacnOJlO>KeHbI TOIJKH K H L TaK, qTO M KIll,
KL 22, LN 101. 4eMY MO>KeT 6bITb p3BHa lIJIHHa oTpe3Ka MN?

KOHTpOJlbHaH pa60Ta
25
2. Ha oTpe3Ke AB pacnOJlO>KeHbI TOqKH C H D TaK, qTO TOqK3 D J1e-
}KHT BHyrpH oTpe3Ka BC. TOlJKH M H N cepeJ{HHbl oTpe3KoB AC It
BD. HaHJlHTe JJ)lHHY oTpe3Ka MN, eCJJH AB 34, CD 12.
3. BHyrpH pa3BepHyroro yrJIa AOB npOBel{eHbl JlytlH DC H OD TaK,
LITO JIyq OC J1e)l{MT BH}'TpH yr J1a BOD. JIY4H 0 L Ii OM HBJ1HIOTCH
6HcceKTpHcaMH yrJIOB AOC H BOD. HaHllHTe rpanycHYIO Mepy yrJl3
COD, eCJIH LMOL 35°.
4. BHYTPH pa3BepH}'TOrO yrJ13 AOB npOBeJleHbl nY1IH OC " OD T3K,
liTO LCOD 81 0 , a rpaJlycHbIe Mepbl yrJloB BOD Ii AOC OTHOCHTCSI
l<aK 2 : 7 COOTBeTCTBeHHO. 4eMY MO)l{eT 6blTb paBH3 rpa.nycHaH Mepa
yrJIa BOD?
5. HeCKOJIbKO npSlMbIx npOBeJIH TaK, liTO OHM pa3eJlHJI" nJlOCKOCTb H3
:()15 113CTeH. KaKoe HaHMeHbllJee 4HCJlO npHMbIX Morno 6bITb npOBeJ!e-
II () ?

26
2. TeOpeMbJ 0 nape napaJ/J1e.nbHbIX npHMblX H CeK)'lllei1
 2. TeopeMblo nape napaJIJleJlbHblX npHMblX H ceKyw.eH
B 3aJlaQaX 1-7 CJIenylOl.l(afi qaCTb YCJIOBHSI HBJlfleTCH o6111ei1:
"Ha pHcYHKe H306pa>KeHbl napaJIJleJ1bHble npHMble a 11 b, nepeCeqeH-
Hble npSlMoH c, H OTMelJeHbI yrJlbl a Ii {3."
I. Hali,UHTe rpa.nycHYIO Mepy yrJl3 ex, eCJ1H yroJl {3 paBeH 30°.
b
c
a
2. HaHltHTe rpaJlYCHYIO Mepy yrJ13 {3, eCJIH yroJl Q paBeH 47°.
b
a
3. HaHllHTe rpa.aYCHYIO Mepy yrJla {3, eC.flH yroJl a paseH 60°.
b
a
4. HaHJ1HTe rpaJlycHYIO Mepy yrJla a, eCJIH OH 60JlblIJe yrJI3 {3 pOBHO
qeTblpe pa3a.
b
a

2. TeOpeMbJ 0 nape napa.n.ne.nbHbIX npHMblX H ceKYllleli
27
5. HaHllHTe rpaJlYCHYIO Mepy yrJ13 {3, eCJIH OH MeHbwe yrJla a pOSHO B
Tpl1 pa3a.
b
c
a
6. HaH!lHTe rpa.nYCHYIO Mepy yr JIa a ecnH pa3HOCTb rpallYCHblX Mep
yr'J10B {3 Ii a paBHa 50°.
b
a
7. HaHDJ1Te rpa.nycHYIO Mepy yrJla a, eCJIH pa3HocTb rpallycHbIX Mep
yrJIOB Q H (3 pasHa 90°.
b
a
B 3allaQaX 8 -II c.neD.YI01ll3S1 43CTb YCJIOBMSI S1BJ1S1eTCH o6111ei1:
"II;) pHCYHKe H306pa)l(eHbi JlBe napbI napaJlJleJIbHblX npS1MbIX: a, b H
- "I, a TalOKe OTMelJeHbI yrJlbl a H {3."
N. IlclHJtHTe rpa,nYCHYIO Mepy yrJla {3, eCJlH yroJl a paBeH 36°.
b
a

28
.
2. TeOpeMbl 0 nape napaJlJ1e.J1bHblX npRMblX H ceKymeii
9. Hai1.QHTe rpaycHYIO Mepy yrJl3 a, eCJlI1 CYMMa rpanycHbIx Mep yr-
JlOB Q Ii {3 paBHa 100 0 .
b
a
1 O. HaH.QHTe rpa.nycHYIO Mepy yrJ1a {3, eCJIH pa3HocTb rpal1ycHblX Mep
yrJ10B Q H {3 pasHa 70°.
b
a
11. HaH,llHTe rpallycHYIO Mepy yr J13 a, eCJ111 OH MeHbwe yr JIa {3 B 3,5
pa3a.
b
a
12. HaHlll1Te rpallYCHYIO Mepy yrJI3 {3, eCJIH OH MeHbwe yrJla Q
B nHTb pa3.
a
b
13. Ha ,naHHOM HH>Ke pHCYHKe H306pa>KeHbl napaJIJIeJ1bHbIe ' '
a H b H nepeceKalOIUHecH npHMble C H d, TOlIK3 nepeCelJeHHfI
Jle>KHT Ha np5!MOH b, a T8K>Ke OTMelleHbl yrJ1bl a, {3 Ii "I. HaHJlHTe -
D.YCHYIO Mepy yrJIa " eCJlH rpaJlYCHble Mepbl yrJ10B Q H {3 paBHbI 30 0

2. TeOpeMbl 0 nape napa.nJ1eJ1bHbIX npHMblX H ceKyweii
50 0 COOTBerCTBeHHO.
29
a
b
14.* lloKa>KHTe, LITO CYMM3 Bcex BHyrpeHHHx yrJIOB JII06oro rpeyroJlb-
liHKa paBHa 180 0 .
15. lloKa)KHTe, liTO eCJlH npHMbIe, co.nep)KalllHe CTOpOHbl ODJ-loro yr J1a,
IlapaJlJIeJ1bHbl npHMblM, COllep/KaIUHM CTOpOHbl !lpyroro yr.na, TO JIH60
'-JTli llB3 yrJla paBHbl, J1H60 HX CYMM3 paSHa 180 0 .
16.* D.oK3)KHTe, liTO eCJlH npSIMble, COJlep}l(3WJte CTOpOHbl OllHoro yrJl3,
IlcpneHHHKYJ151pHbl npHMblM, COJlep>KalllHM CTOpOHbl Jlpyroro yrJI3, TO
.111160 3TH D.B3 yrJ1a paBHbl, JIH60 HX CYMMa paSHa 180 0 .
17. bHcceKTpHcbl Bcex BHyTpeHI1Hx yrJ10B napaJIJIeJlOrpaMM3 ABCD
11J)()llOJl>KHJlH 110 nepecetleHllSi B TOIIKax 1(, L, M, N, CM. JlaHHblM HH)Ke
I" ICYHOK. D.oKa/KHTe, liTO IJeTblpexyrOJIbHHK KLJ\;[N HB.nHeTCSI npHMO-
\,1 ()L1JblHKOM (T.e. Bee ero yrJIbl npHMbIe).
B
c
A
D

30
.
3. TpeyrOJIbHJfK
3. TpeyroJlbHltK
3.1. npH3HaKH paBeHCTBa TpeyrOJ1bHHKOB.
3aAaLJH Ha AOKa3aTeJlbCTBO
B 3aJla l iax NQ 1- 5 HCnOJIb3YHTe nepBblH npH3HaK paBeHcTB3 Tpe-
yrOJIbHHKOB.
1. a) lloKa)KHTe lITO B paBHo6ellpeHHOM TpeyroJlbHHKe 6HcceKTpHca
..
yrJla, 3aKJII04el-il-iOrO Me>KllY paBHbIMH CTOpOH3MH, HBJ1HerCH T3Kme 'e-
u u
lIJiaHOH H BblCOTOH 3Toro TpeyroJlbHHKa.
6) lloKa)KHTe, liTO B paBHo6en.peliHOM rpeyroJlbHHKe yrJ1bl npH Oc-
HOBal1HH paBHbl.
2. ,UOK3>KI1Te, liTO B pasHo6ellpeHHOM TpeyrOJJbJtHKe Me.QHaHbl, npOBe-
..
lleIHble 113 BepWI1H OCHOBal-IH5I, paBI-Ibl.
A
B
L
3. Ha CTopOHax paBIOCTopOHHero Tpeyro.nbHHKa
ABC OT J10>KeHbl paBHble oTpe3KH AK, B L, C /\1,
K3K nOK33aHO Ha pHCYHKe. HOKa>KHTe, 4TO rpe-
yrOJIbHHK K L1\.[ S1BJI5JeTCSI paBHocTopOHHHM.
c
4. CropoHbI paBliocTopoHHero TpeyrOJ1bHHKa ABC
npOl!OJI>Kellbl H8 paBHble oTpe3KH AK, BL H CN!
T3K, KaK nOK333HO H3 pHcYHKe. lloKa)KHTe, IjTO
TpeyroJlbHHK K LA! HBJISleTC5I paBHocTopOHHHM.
5. TOllKH J.11, N, P - cepeAHHbl CTOpOH paBHo6ellpeHIIoro TpeyrOJIbHH-
Ka ABC. lloKa)KHTe, 4TO Tpeyro.nbHliK Af N P - paBH06eJlpeHHbIH.
B 3aAal.faX N2 6 - 8 H NQ 10 licnoJlb3YHTe BTOpOH npH3HaK
TpeyrOJIbHHKOB.
6. HOKa>KHTe, 4TO B paBHo6e,QpeHHoM TpeyrOJ1bHHKe 6HcceKTpHcbl -
JlOB npl1 OCHOS3HHH paBHbl.

ITpH3HaKH paBeHCTBa TpeyrO.RbHHKOB. 3a,aaQH H8,O.OKa3aTeJ1bCTBO 31
7. l{oK3)KHTe, IJTO npoTHBOnOJIO}KHble CTOpOHbl napaJlneJIOrpaMM3
pa BHbl.
8. l{oK3>KHTe, IJTO }!HarOH3JIH napaJlJ1eJ10rpaMMa TOqKOH nepeCeqeHHH
,QeJIHTCH nOnOJlaM.
9. ,UOKa>KHTe, liTO eCJlH .nHarOHaJlH lJeTbIpexyroJlbHHK3 TOtJKOH nepe-
..
cetleHHH lleJ1HTCH nOnOJlaM, TO 3TOT 4eTblpexyroJlbHHK - napaJlJIeJlO-
rpaMM.
1 O. ,UOKa>KMTe, 4TO eCJlH JlBe npOTHBOnOJlO>KHble CTOpOHbl lJeTbIpex-
..
yrOJIbHHKa napaJlJIeJlbHbl Ii paBHbl ltpyr JlPYry, TO 3TOT qeTbIpexyrOJIb-
IIHK - napanJleJIOrpaMM.
B 3a,QalJ3X NQ 11-13 HCnOJIb3YHTe TpeTHH npH3HaK paBeHcTB3 Tpe-
yrOJIbHHKOB.
11. l{oKa>KHTe, 4TO eCJIH npoTHBOnOJIO)KHble CTOpOHbl lJeTblpexyrOJIb-
..
11111<3 nonapHO paBHbl, TO 3TOT qeTblpexyroJlbHMK napaJlneJIOrpaMM.
12. lloKa>KHTe, qTO eCJlH llHarOHaJlH napaJlJIeJIOrpaMMa paBHbl, TO 3TOT
I I; I r)3JI.fleJIOrpaMM HBJISIeTCSI npHMoyrOJIbHHKOM.
I :J. ]JOKa>KHTe, liTO ,llliarOHaJlH pOM6a nepneHllHKYJlSlpHbI.
I /1. ]JOK3>KHTe, liTO JlMarOHaJiH npHMoyroJlbHHKa paBHbl.
I fJ.* 'uOKa>KHTe, liTO B npSIMoyroJlbHoM rpeyronbHHKe J]JIMH3 MellHaHbl,
.. "
1'1 )()I\CD-eHHOH K rHnOTeHY3e, paBHa no.nOBHHe JlJIHHbl rHnOTeHY3bI.
I ti: k 'IOIIKH M H N cepeAHHbl CTOpOH AB H AC TpeyroJlbHHKa ABC.
, C ) II )l:iOK M N H33blBaerCH CpeJlHeH JIHHHeH TpeyroJlbHHKa ABC.)
I/;k l'ollKH M H N cepellHHbl OOKOBbIX CTOpOH AB H C D Tpane-
'11111 /lBCD. HOK3>KHTe, liTO OTpe30K MN napaJIJleJleH OCHOBaHHHM
· I.t 'lilt Iii J1HHHeH TpaneUHH).

32
.
3. TpeyroJlbHHK
3.2. CYMMa yrJloB TpeyroJlbHHKa
18. B TpeyroJlbHHKe ABC yroJl A paBeH 24°, BHelllHHH yroJ1 np" Bep-
llJHHe B paBeH 123 0 . HaHllHTe rpaJlycHYIO Mepy yrJla C.
19. B TpeyrOJIbHHKe ABC BHeWHHe yrJlbI npH BepwHHax B H C paB-
Hbl 91 0 Ii 99° COOTBeTCTBeHHO. HaHJlHTe rpanycHYIO Mepy yrJl(J A.
20. B rpeyrOJ1bHHKe ABC yrOJ1 C paBeH 50°, AC BC. HaHl1HTe
rpaJ1ycHYIO Mepy BHelllHero yr JIa npH sepwHHe B.
21. B TpeyroJlbHHKe ABC BHeWHHH yroJI npH BepWI1He B paBeH
124°, AC BC. HaHJlHTe rpaJlycHYIO Mepy yrJla C.
22. rpalI.ycHble Mepbl yrJIOB TpeyrOJIbHHKa OTHOCHTCSI KaK 3 : 5 : 7.
HaHltHTe rpaJlycHYIO Mepy MeHblllerO 113 yr.nOB TpeyrOJ1bHHKa.
23. rpanycHble Mepbl yrJloB TpeyrOJIbHHKa OTHOC5ITC5I K3K 1 : 2 : 3.
HaHJlMTe rpallycHYIO Mepy 60J1bWerO 113 yr JlOB TpeyrOJIbHHKa.
24. O!tHH 113 BHelllHHX yrJloB TpeyrOJIbHHK3 paBeH 143 0 .
Mepbl yrJloB TpeyrOJIbHHK3, He CMe)l{HblX C JlaHHblM BHeWHHM yrJIOM,
HOCHTCH KaK 4 : 9. HaHHHTe rpaJlycHYIO Mepy MeHbwero H3 3THX
25. OJlHH 113 BHCWHHX yrJIOB TpeyroJlbHHKa paBeH 132 0 . 
Mepbl yrJloB TpeyrOJIbHHKa, He CMe)l{HblX C llaHHbIM BHeWHHM yrJlOM,
HOCSITCH K3K 5 : 6. HaHJlHTe rpaJ1ycHYIO Mepy 60JIblllerO H3 3TMX yr
-
26. B TpeyroJlbHHKe ABC CYMM3 BHelllHero yr.na npM sepllIHHe A
BHYTpeHHHx yr.nOB np" BepwHHax B M C Ii paBH3 156 0 . HaH,QHTe rpa-
.nYCHYIO Mepy BHyrpeHHero yrJIa npH BeplllHHe A.
27.* B TpeyroJlbHHKe ABC CYMMa BHYTpeHHero yr JIa npH BepllIl1He
Ii BHelllHHX yrJIOB npH BeplllHHax B Ii C paSHa 196 0 . Hati,QHTe
HYIO Mepy BHyrpeHHero yrJIa npH BepwHHe A.
28. O,aHH H3 BHelllHHX yrJIOB paBH06eJlpeHHOro TpeyrOJIbHHKa
130 0 . HaHllHTe rpaJlycHYIO Mepy 60JIbllJerO 113 BHYTpeHHHX
TpeyrOJIbHHKa.

CYMM8 yrJ/OB TpeyrOJlbHIIKa
33
29. OllHH H3 BHeWHHX yrJloB paBHo6e.npeHHoro TpeyrOJIbHHKa paBeH
1100. l-IaHHHTe rpaJlycHYIO Mepy MeHbwero H3 BHYTpeHHHx yr JlOB 3Toro
TpeyroJlbHHKa.
30. B TpeyrOJIbHHKe ABC yrOJI C paseH 77°, AD 6HcceKTpMca yrJIa
A, yroJl BAD paBeH 23 0 . HaHJ1HTe rpaJJ}'cHYIO Mepy yrn3 BDA.
31. B TpeyrOJIbHHKe ABC yroJl C paBeH 44°, AD 6HcceKTpHca yrJla
A, yron ADC paBeH 100 0 . HaHLlHTe rpaJlycHYIO Mepy yrJIa B.
32. B TpeyrOJIbHHKe ABC yroJl C paBeH 25°, AD 6HcceKTpHca yrJla
i1, yroJl ADB paBeH 80°. HaHJ1HTe rpa,nycHYIO Mepy yrna B.
:33. B TpeyrOJIbHHKe ABC yroJl B paBeH 75°, AD 6HcceKTpHca yr-
.rIel A, yroJl C MeHbwe yrJI3 ADB B BoceMb pa3. HaHllHTe rpallycHYIO
t\1l'PY yrJIa C.
:J4. B TpeyroJlbHHKe ABC yroJl C paBeH 40°, AD - oHcceKTpHca yrJIa
:\, yroJl B 6o.nbwe yr.na ADB B llB3 pa33. HaHHTe rpan.ycHYIO Mepy
VI )la B.
:15. B TpeyrOJIbHHKe ABC yrOJI C paBeH 124°, 6HcceKTpHcbl yrJloB A
II /111epeceKaIOTc5I B TOtlKe O. HaHllHTe rpaltYCHYfO Mepy yrJIa AOB.
:It). B TpeyrOJIbHHKe ABC 6HcceKTpHcbl yrJloB A H B nepeceK310TCH
U I ()IIKe 0, yroJl AOB paBeH 129°. HaH1!HTe rpallycHYIO Mepy yrJIa C.
:1'7. B TpeyrOJIbHHKe ABC yroJ1 C paBeH 30°, 6HcceKTpHcbI BHew-
lilt \ yrJ10B npH BepwHHax A H B nepeceK310TcH B TOIIKe P. HaHD-MTe
II t.llIYCIYIO Mepy yr J13 AP B.
:11-\. 13 rpeyrOJIbHHKe ABC 6HcceKTpHcbl BHeWHHX yrJIOB npH BepwH-
lI.t' /1 Ii B nepeceKBIOTCS1 BTOllKe P, yron APB paBeH 31°. HaHJlI1Te
Ilt.IIIY,:IIYIO Mepy yr J13 C.
.II. 1\ ()CTpoyrOJ1bHOM Tpeyro.nbHHKe ABC BblCOTbl AAI H BBI 06-
" , 'Vie"" CO CTOpOHOH AB yrJIbl B 34° H 43° COOTBeTCTBeHHO. HaHD.HTe
· I,.IIIY( 'IIYIO Mepy yr JIa C.
II' , K "I' 11o.nr K Or). TeM TCCThl

34
.
3. TpeyrOJIbHHK
40. B rpeyroJlbHHKe ABC yroJI C TynoH. BblCOTbI AAI H BBI 06-
pa3YlOT co CTOpOHOH AB yr.nbI B 45° H 60° COOTBeTCTBeHHO. HaHllHTe
rpallycHYIO Mepy yr JIa C.
41. B TpeyroJlbHHKe ABC yroJl B TynoH. BbICOTbl AAI H BBI 06-
pa3YlOT co CTOpOHOH AB yrJIbl B 30° H 50° COOTBeTCTBeHHO. HaHJlHTe
rpallycHYIO Mepy yr Jla C.
42. B rpeyroJlbHHKe ABC yrOJI A paBeH 30°, yroJl B paBeH 40°.
npHMbIe, cOJlep)l{alllHe BblCOTbl AAI Ii BB l , nepeceK3lOTC5I B TOlJKe H.
HaH,QHTe rpallycHYIO Mepy yrna AH B.
43. B TpeyrOJIbHHKe ABC yroJl A paBeH 50°, yroJI B paBeH 100 0 .
npHMble. cOJlep}l{alllHe BbICOTbI AAI H BBI' nepeceKalOTC5I B TOllKe H.
HaHJ1HTe rpaHycHYIO Mepy yrJIa AHB.
44. B TpeyroJlbHHKe ABC yroJl B paBeH 45°, yroJ1 C paseH 75°.
HaHllHTe rpallycHYIO Mepy yrJla Me)KllY BblCOTOH H 6HcceKTpHcoH, Bbl-
XOllHlllHMH 113 BepWHHbl A.
45. B TpeyrOJIbHHKe ABC yrOJI B paBeH 20°, yroJI C paseH 120°.
HaHJ!HTe rpaJlycHYIO Mepy yrJ1a Me)l{JlY BblCOTOH H 6HcceKTpHcoH, Bbl-
XOMllllHMH H3 BepwHHbl A.
46. Ha llaHHOM cnpaBa pHcYHKe yroJl 1 paBeH
30°, yroJl 2 paBeH 20°, yroJ] 3 paBeH 45°. Hai1-
LlHTe rpaJlycHYIO Mepy yr JI8 4.
47. Ha LlaHHOM cnpaBa pHcYHKe yroJ1 1 paBeH
40°, yroJl 2 paBeH 87°, yron 4 paBeH 120°. HaH-
,LlHTe rpaJIycHYIO Mepy yrJIa 3.
48. Ha JtaHHOM cnpaS3 pHCYHKe yroJJ 2 paBeH
113°, yroJl 3 paBeH 131 0, yron 4 paBeH 1300. HaH-
D-HTe rpan.ycHYIO Mepy yr JIa 1.

17'JRAloyroJlbHblH TpeyroJlbHHK
35
49. Ha llaHHOM cnpaBa pHcYHKe yroJil paBeH 35°,
yrOJ1 2 paBeH 140°, yroJl 3 paBel-l 135°. HaH!lHTe
rl)allycl-IYIO Mepy yr JIa 4.
50. fl.oKa)K11Te, liTO CYMMa BCeX BHyrpel1HHX yrJ10B .n106oro lJeTblpeX-
yrOJlhHHKa paBI1a 360 0 .
51. fl.oK3)KHTe, LITO eCJJI1 npOTHBOnO.JIO)KHble yrJIbI lleTbIpexyroJlbHHKa
..
11()llapI-IO paRI-Ibl, TO 3TOT 4eTblpexyrO.nbHl1K SlBJ1HeTCH napaJlJIeJlOrpaM-
1()M.
52.* 6licceKTpHcbI BHyrpeHHHx yr.noB rpeyro.nbl-lHKa
.\ 13C npH BepwHHax A 11 B n epeCeK alO TC5I B TOll-
B
A
!):J. bllCCeKTpHCbl BI1elllHHX yr.nOB rpeyrO.nbHl1K3
\ I C 11PH BepUJHlaX A H B ne peceKal O TCSI B TOll-
3.3. npSlMoyroJlbHblH TpeyroJlbHHK
!,.I. ()JtHH f3 OCTpblX yrJIOB npHMoyroJlbHoro TpeyroJlblHKa paBeH 17°.
11.11'1 '1.I1Te rpaJtycHYIO Mepy yr JIa Me>K.n.y BblCOTOH Ii 6HcceKTpMcoH, npo-
lit '11'11I11)IMH H3 BepwHHbl npHMoro yrJl3.
I:'. I Ilp5IMoyroJlbHoM TpeyrOJIbHHKe yroJl Me>Kn.y BblCOTOH H 6HcceK-
. t till '( )i'I, npOBeJ1.eHHbIMH H3 BeplllHHbl np5lMOrO yrJIa, paBeH 12°. HaHHH-
II II );IJLYCI-IYIO Mepy MeHbllJerO yrJIa 3Toro TpeyrOJIbHHK3.
I  l}lIIJeHHH 3allaLJ N2 56 - 62 HcnoJlb3YMTe pe3YJlbTaT 3allaIJH Ng15
t . ,. I (f!" aKT, 4TO Me.nHaHa K rHnOTeHY3e npHMoyrOJIbHOrO TpeyroJlb-
.. tll " I ,; 11lla nOJIOBHHe fMnOTeHY3bl).

36
3. TpeyrOJ/bHHK
56. OHHH 113 OCTpblX yrJ10B npHMoyrOJ1bHOrO TpeyrOJIbHHKa paBeH 50°.
HaHJlHTe BeJ1H l IMHY MeHbwero H3 yrJlOB, o6pa30BaHHbix K3TeTaMM 3Toro
U .. U
rpeyrOJIbHHKa H MejlH3HOH, npoBe.neHHoH K rHnoTeHY3e.
57. On.HH H3 OCTpblX yrJ10B npSIMoyrOJ1bHOrO TpeyrOJIbHHKa paBeH 49°.
Haii.aHTe BeJlHlJHHY 60JlbWerO H3 yrJ10B, o6pa30BaHHbix KaTeT3MH 3Toro
U .. U
TpeyrOJ1bHHKa H MellHaHOH, npoBelleHHoH K rHnOTeHY3e.
58. OllHH H3 OCTpblX yrJloB npSlMoyrOJ1bHOrO TpeyrOJlbHHKa paBeH 48°.
HaHJlHTe rpallYCI-IYIO Mepy yr J1a Me>KllY BblCOTOH H MellH3HoH, npoBe-
..
lleHHblMH H3 Bepl.llHHbl npSlMoro yrJla.
59. B npSlMoyrOJIbl-lOM TpeyrOJ1bHHKe yrOJ1 Me*n.y BblCOTOH H MeltHa-
HOH, npoBe,neHHblMH H3 BepWHHbI npSIMoro yrJI3, paBeH 20°.
rpallycHYIO Mepy 60JIbWerO H3 OCTpblX yrJloB 3Toro TpeyroJlbHHKa.
60. OllHH 113 OCTpblX yrJ10B npHMoyrOJlbHOrO TpeyrOJ1bHHKa paBeH 72°
HaHJlHTe rpaJtycHYJO Mepy yrJIa Me>KllY 6HcceKTpHcoH H MellHaHoH,
..
Be.aeHHbIMH 113 BepWHHbl npSlMOrO Y'''JIa.
61. B np5lMoyrOJIbHOM TpeyroJlbHHKe yrC)J] MelK,IlY 6HcceKTpHcoH Ii
Jl.Hal-IOH, npOBelteHl-lblMH 113 BepWHllbl flpHMOrO YIJl3, paBeH 24°.
Te rpa.uycl-lYlo Mepy 60JIbWeI"o li3 OCTpblX yrJloB 3Toro TpeyrOJ1bHHKa.
62. llOKa>KHTe, 4TO B npHMoyrOJIbHOM rpeyrOJIbHHKe 6HcceKTpHca
u u
npHMoro yr JI3 JleJJHT nOnOJIaM yroJi MeJKllY Me,[lHaHOH H BbICOTOH,
.. U
npOBeJleHHblMH H3 3TOH )Ke BepWHHbI.
63. B npHMoyrOJ1bHOM TpeyrOJIbHHKe ABC c rHnoTcHY30H AB n
BelleH3 BblCOTa CH, oTpe30K CD 6HcceKTpHca Tpeyr0J1bHHK3 AC
llOKa>KMTe, qTO BD BC.
64.* llaHa TpaneLlHSI C OCHOB3HHHMI.{ AD Ii BC. 5HcceKTpHcbl
peHHHx yrJ10B npH BepWHHax A H B nepeceKalOTCH B T04Ke P, a I
ceKTpHCbl BHYTpeHHHX yr JlOB np" BepWHHax C H D 1
TOqKe Q, CM. J{3HHbIH HH>Ke PHCYHOK. ll O K a >KHTe, 4TO orpe30K PQ

HepaBeHCTBO TpeyrO.nbHHK8
37
A
65.* llaHa TpaneLlHH C OCHOBaHHMH AD H BC. DHcceKTpHcbl BHew-
11HX yrJloB np" BepwHHax A H B nepeceKalOTCH B T04Ke P, a 6HcceK-
'rpl1Cbl BHeWHHX yrJ10B npH BepWHHax C H D nepeceKalOTCfI B TOllKe
(J, CM. 113HHbIH HH)I{e PHCYHOK. llOK3>KHTe, 4TO OTpe30K PQ napaJlJ1e-
.1ICH OCHosaHHS1M rpaneLlHH, a ero .ltJIHH3 pasHa n0J10BHHe nepHMeTpa
-rpaneUHlt ABC D.
B
c
A
D
66.* llaHa TpaneUHSI TpaneUHSI ABCD C OCHOB3HHSlMH AD H BC,
I ;11<aH, lITO AD > BC, a CYMl\13 ee BHYTpeHHlix yrJloB np" OCHOBaHHH
\ fJ pasHa 90°. ,QOK8)1{HTe, lITO J]JIHH3 O T pe3Ka, coellHHSIlOll.lerO cepe-
'.7.* naH paBHo6e,npeHHblH TpeyroJlbHHK ABC C OCHOB31-1HeM AC,
· t II )C30K C L ero 6HcceKTpHca. nepneHllHKYJ1Hp K C L, npoxo,nSlLllHH
'I' 'I )l\3 T04KY L, nepeceKaeT npS1MYIO AC B T04Ke [(. ,QOKa)l{HTe, liTO
, '/\' == 2AL.
3.4. HepaBeHcTBo TpeyroJlbHHKa
tiS. nepHMeTp paBH06eJlpeHHOro Tpeyro.nbHMKa paBeH 57, a O,l!H3 H3
, I . . "'r()pOH paBHa 27. HatiD-HTe CTOpOHbl 3Toro TpeyrOJIbHHKa. B OTBeTe
,-I .lil\lrre MeHbwylO H3 CTOpOH.
fll, I IepHMeTp paBHo6ellpeHHoro TpeyronbHHKa paBeH 88, a Ol!Ha H3
· , , ., I ()I)OH pasHa 20. HaHJlMTe CTOpOHbl 3Toro TpeyroJlbHHKa. B OTBeTe
I' .. ,h 1ft C 60JlbWYlO H3 CTOpOH.
itl 11(\pHMeTp paBH06eJIpeHHOro TpeyrOJIbHHKa paBeH 34, a pa3HocTb
... I .) I. )I)I)IX nBYX ero CTOPOH pasHa 10. HaHJ!HTe JI..nHHY OCHOBaHHSI 3TO-
I , I' Ie 0VI'()JJ bHHKa.

38
3. IjJeyroJIbHHK
71. nepHMeTp paBH06eJ1peHHOro TpeyrOJIbHHKa paBeH 50, a pa3HocTb
HeKoTopblX llBYX ero CTOpOH pasHa 13. HaHD-HTe llJlHHY 60KOBOH CTO-
pOHbl 3Toro Tpeyro.JlbHHK3.
72.* OTpe30K C AI - Me.nHaHa TpeyronbHHK3 ABC. llOK3)KHTe, liTO:
a) eCJ11t C AI AB /2, TO yroJi C npHMoti;
6) eCJ]i C 1\[ < .l4B /2, TO yro.n C Tynoi1;
B) eCJlH C AI > AB /2, TO yrOJ1 C OCTpbIH.
73.* -Use CTOpOHbl Tpeyro.nbHHK3 He paBIbl1!pyr APYry. nOKa>KHTe, liTO
Men.l1aHa, npOBelI.eHHa51 113 HX 06meH BeplllHHbl, o6pa3yeT C 60J1bLUeii H3
u
3THX CTOpOH MeHbWHH yroJl.
74.-1: OTpC30K Bltl - MeJIJlaHa TpeyrOJ1bHHKa ABC. nOKaiKH1e, LITO
CnpaBeJlJIHIbI C Jl ellYIOllillC HepaBel-ICTBa:
75.* Ha nlIOCKOCTH 3a1l8Hbi npHMoyro.nbHHK ABC D H T04Ka AI.
K(l)l(HTe, (ITO AAI < BAI + CA,! + DPvI, BHe 3aBHCliMOCTH OT
TOlIKI1 1\1.
76.* llall rpeyroJlbHHK ABC Ii TO(IK3 AI, JleiKallla51 BHYTPH Hero.
Ka>KHTe, liTO cnpaBen.J1HBbl CJ1enylOmHe HepaBeHCTBa:
a) Al\J + Bl\l < AC + BC;
6) AAI + Bl'l + CPvI < AB + BC + AC.
77.* llOKa)l{HTe, tlTO B .n1060M BblnYKnOM LJeTbipexyroJlbl1HKe
D.J1HH AHarOHaJleH Melbwe nepHMeTpa, HO 60JIbWe IloJlynepHMeTpa.
78.* llOKa}J{HTe, lITO B J11060M BhlnYK.nOM n5lTHyr"OJ1bHHKe CYMM3 '
Bcex llHarOH3JIeH 60J1bWe nepHMeTpa, HO MeHbwe Yl1BOeHHOro :}
MeTpa.

KOHTpO.!lbHan pa60ra no TeMaM 2 H 93
39
KOHTpOJlbHaSi pa60Ta no TeMaM 2 H 3
PeKoMeHnyeMoe BpeMH .QJJH BbInOJlHeHHH pa60TbI 40 MHHyr.
PeKOMeHJlYMaSl WKaJI3 OLleHHB3HHSI pe3YJIbT3TOB:
.... 0-1 2-3 4 5
'-IlI1cno BepHblX peweHIIIM
WKOJ1bHa ou.eHKa 2 3 4 5
BapHaHT 1
1. 8 TpeyronbHHKe ABC yro.n A paBeH 11 0, BHelllHHH yroJl npM Bep-
III HI-Ie B paseH 101 0 . HaHlll1Te rpaJIycHYIO Mepy yrJla C.
2. 8 TpeyroJlbHHKe ABC yroJ1 C paseH 72°, AD - 6HcceKTpHca yrna
· \, yrOJ1 BAD paBeH 27°. HaH,QHTe rpanycHYIO Mepy yrJI3 BDA.
:1. I-Ia pHcYHKe H306pa>KeHbI napa.nJIeJIbHble npSlMbIe a Ii b, nepece-
tlt"llllbIe npHMoH c, H OTMelleHbl yrJlbl a H (3. HaH!lHTe rpaJlycHYIO Mepy
VI.'I,) {3, eCJIH yroJl Q paBeH 130 0 .
b
c
a
I. ()JIHH H3 OCTpbIX yrJloB np5lMoyroJlbHoro TpeyrOJ1bHHKa paBeH 46°.
11.II'IJll-ITe rpaJlycHYIO Mepy yrJIa Me}l(llY BblCOTOH H Me,UH3HoH, npoBe-
I'.IIIIIIMH 113 BepWHHbI npHMoro yr.na.
, II.( )Ka)KHTe, 4TO B paBHo6ellpeHHOM TpeyroJlbHHKe BbICOTbl, npOBe-
,. 11I1I.le K paBHbIM CTopOHaM, paBHbI.
BapHaHT 2
I I \ II U..'y,-'o.nbHHKe ABC BHelllHHe yrJIbI npH BeplllHHax B Ii C paBHbI
't I II )!)O COOTBeTCTBeHHO. HaHJtHTe rpaJlycHYIO Mepy yrJIa A.

40
.
KOHTpOJlbH8n pa60r8 no TeM8M 9 2 II 93
2. B TpeyroJlbHMKe ABC yroJi C paBeH 55°, AD 6HcceKTpHca yrJl3
A, yroJl ADC paBeH 110 0 . HaHllHTe rpaJlycHYIO Mepy yrJIa B.
3. Ha pHCYHKe H306pa>KeHbl napaJ1J1eJ1bHble npHMble a H b, nepece-
lJeHHbIe npHMoM c, Ii OTMelleHbl yrJIbI Q H {3. HaHllHTe rpan.YCHYIO Mepy
yrJla 0, ec.nH yroJl {J paBeH 110 0 .
b
a
4. OllHH H3 ocrpbIx yrJIOB npS1MoyrOJIbHOrO Tpeyr0J1bHl1K3 paBeH 89°.
HaHJ{HTe rpa.nycHYIO Mepy yr.na Me)KllY 6HcceKTpHcoH Ii 1\1eJlHaHOH, npo-
..
BeneHHbIMH H3 BepwHHbI npHMoro yr.na.
5. ,lIoK3>KHTe, qTO eCJIH !tBe BbICOTbl TpeyrOJIbHHKa pasHbl llpyr ltpyry,
TO CTOpOHbl, K KOTOpbIM npOBe,lleHbl 3TH BbICOTbl, TalOKe paBHbl.
BapHaHT 3
1. B TpeyrOJIbHHKe ABC yroJl C paBeH 70°, AC BO. l-1aHllHTe rpa-
nyCHYIO Mepy BHeWHero yrJla npH BepwHHe B.
2. B TpeyroJlbHHKe ABC yroJI C paBeH 20°, AD - 6HcceKTpHca yrJIa
A, yrOJI ADB paBeH 60°. HaHJI.HTe rpallycHYIO Mepy yrJ1a B.
3. Ha pHcYHKe H306pa)KeHbl napaJlJleJIbHble npHMble a Ii b, nepece-
qeHHbIe npSIMoH c, H OTMe4eHbI yrJlbl Q H {3. HaMJtHTe rpanycHYIO Mepy
yrJ1a a, eCJIH H3BeCTHO, liTO OH 60JIbLUe yrJla {J B 2,6 pa3a.
b
a

I(OHTpOJ/bHan pa60T8 no TeM8M 2 H 3
41
4. B npSIMoyrOJIbHOM TpeyrOJIbHHKe yroJl Me)l{JlY BblCOTOH " MeD.HBHOH,
npOBelleHHblMH 113 BepWHHbI npHMoro yrJIa. paBeH 10°. HaHJlHTe rpa-
IlYCHYIO Mepy 60JIbWerO H3 OCTpblX yrJloB 3Toro TpeyrOJIbHHK3.
5. llOK3)1{HTe, 4TO eCJIH J1HarOH3J1H napaJlJIeJIOrpaMMa nepneHD-HKYJlSIp-
1-lbI, TO T3KOH napaJl.neJIOrpaMM HBJlHeTCH pOM6oM.
BapHaHT 4
1. B TpeyrOJIbHHKe ABC BHeWHHH yroJl np" nepwHHe B paBeH 111 0,
44C BC. HaH.QHTe rpaAycHYfO Mepy yrJIa C.
2. B Tpeyro.nbHHKe ABC yrOJ1 C paBel-t 35°, AD 6HcceKTpHca yrJIa
/1, yroJl ADB paBeH 95°. HaHlIJtTe rpaJIycHYIO Mepy yrJla B.
:1. Ha PHcYliKe H306pa)l{eHbl napaJlJ1eJ1bHble npHMble a H b, nepece-
'1i'IIHbIe npHMoH c, Ii OTMe4eHbi yrJlbl a " {3. HaHllHTe rpa!lycHYIO Mepy
Yl'JI3 0:, eCJIH H3BeCTHO, liTO OH MeHbwe yr J1a {3 B 1,4 pa33.
b
a
.1. I  IlpHMoyroJlbHoM TpeyrOJIbHHKe yro.n Me)l{llY 6HcceKTpHcoH " MellH-
1111f »ii, npOBe.neHHbIMH 113 BepWHHbI npHMoro yrJIa, paBeH 15°. HaH.llHTe
I I t,l}l.YCI-fYIO Mepy 60.nbllIero 113 OCTpblX yr nOB 3TOrO TpeyroJlbHHKa.
:,. } l( )1<a)l{HTe, tJTO eCnH l{HarOHaJIH pOM6a paBHbl, TO TaKO" pOM6 HBJ1H-
· I. I I KIJan.paTOM.
BapHaHT 5
I II) l.1lYCI-IbIe MepbI yr JlOB TpeyroJlbHHKa OTHOCSITCH KaK 5 : 13 : 18.
, I III '1llre rpa.uycHYIO Mepy MeHbllJerO H3 yr JIOB TpeyrOJIbHHKa.

42
KOHTpOJIbHan pa60r8 no TeMaM 2 H 3
2. B oCTpoyroJlbHoM TpeyrOJIbHHKe ABC BblCOTbI AAI 11 BBI 06pa3Y-
lOT co CTOpOHOH AB yrJlbl B 23° H 32 0 COOTBeTCTBeHHO. HaHJlHTe rpa-
llYCHYIO Mepy yrJ1a C.
3. Ha pHCYHKe H306pa>KeHbI ,llBe napbl napaJI.neJIbHblX npS1MbIX: a, b H
c, d, a ralOKe OTMeqeHbI yrnbl a H {3. HaHLlHTe rpal!YCHYIO Mepy yrJI3 {3,
eCJIH yroJl Q: paBeH 124 0 .
b
a
4. OepHMeTp paBH06eJlpeHHOro TpeyrOJIbHHKa paBeH 31, a Q,llHa 113 ero
CTOpOH paBH3 13. HaHAHTe CTOpOHbl 3Toro rpeyrOJIbHHKa. B OTBeTe
YKa}KHTe MeHblllYIO 113 CTOpOH.
5. B paBHo6eJ!peHHoM TpeyroJlbHHKe ABC C OCHOBaHHeM BC -
JleHbI BbICOTbl BD Ii CE. ,QoKa>KHTe, 4TO npHMble BC H DE
JIeJIbHbl.
BapHaHT 6
1. rpa.nycHble Mepbl yr JIOB TpeyrOJIbHHKa OTHOCHTCH KaK 2 : 3 : 7. Hati
JlHTe rpaJlycHYIO Mepy 60JIbWerO 113 yrJloB TpeyrOJIbHHKa.
2. B TpeyrOJIbHHKe ABC yroJI C Tynoti. BbICOTbI AAI Ii BBI -
pa3YlOT co CTOpOHOH AB yrJIbI B 49° H 51 0 COOTBeTCTBeHHO.
rpaJJycH)'1O Mepy yr JI3 C.
3. Ha pHCYHKe H306pa>KeHbI J{Be napbI napaJlJIeJIbHbIX npHMblX: a, b
c, d, a TalOKe OTMelJeHbI yrJIbl (} H {3. HaHllHTe rpanYCHYIO Mepy yrJIa
eCJI" yron {3 paBeH 144 0 .
b
c
a

KOHTpOJlbH8n pa60ra no reMaM 2 II 3
43
4. nepHMeTp paBH06el!peHHOro TpeyrOJIbHHK3 paBeH 89, a OHHa 113 ero
cropOH paBlia 29. HaHJ1MTe CTOpOHbI 3Toro TpeyroJlbHHK3. B OTBeTe
YKa)KHTe 60JlbWYIO 113 CTOpOH.
5. B paBH06e1!peHHOM TpeyroJlbHHKe ABC" C OCHOB3HHeM BC npo-
relleHbl 6HcceKTpHcbl BK H CL. .aOKa)l{HTe, liTO npSlMble BC H K L
IlapaJIJIeJIbHbl.
BapHaHT 7
1. OLlHH 113 BHeWHHX yrJ10B TpeyroJlbHHKa paBeH 135 0 . rpaJlycHble MC-
()i)1 yrJ10B Tpeyr0J1bHHK3, He CMe)KHbIX C 113HI-IbIM BHeWHHM yrJ10M, OT-
11()CTCSl KaK 7 : 8. HaHJ1HTe rpaJtycHYIO Mepy MeHbwero H3 3THX yr J10B.
. B TpeyrOJIbHHKe ABC yrOJ1 B Tynoti. BblCOTbI AAI 11 BBI 06-
1);I:YIOT co CTOpOHOH AB yrJIbl B 20° H 35° COOTBeTCTBeIHO. Haitlll1Te
II ))JlYCI1YIO Mepy yrJl3 C.
:1. J-Ia pHcYHKe H30Upa}l{CHbl JlBe napbl napaJIJ1eJ1bHblX np5lMblX: it, b H
, ,(I, a T3K)I(e OTMelleHbl yrJlbl a 11 (3. Hai'll1HTe rpallYCHYIO Mepy yrJla (3,
c'c'.1111 li3BeCTHO, lITO OH MeHbllJe yrJI3 Q B 3,8 pa33.
b
a
I 'I <'pHMeTp paBH06el1peHHOro Tpeyr0J1bHHK3 paBeH 76, a CYMM3 HeKO-
14 'I 'I.IX llBYX era CTOpOH paSHa 54. HaHJ!HTe )lJIHHY OCl-IOBaHHH 3Toro
11"  YI ()JI bHHKa.
 I J I.c )I{(I)KHTe, liTO eCJlH MenHaHa H BbICOTa, npOBen-eHHble K fHnOTeHY-
.. 11'111()rO npHMoyroJlbHoro TpeyrOJIbHHKa paBHbl COOTBeTCTBeHHO Me-
..
111.1I1t. 11 BbICOTe, npOBeJI.eHHblM K rHnOTeHY3e Hpyroro npSlMoyrOJIbHOrO
u
tl" \'1 f ).III)I-ll-fKa, TO nepBbIH H3 3THX TpeyrOJIbHHKOB p3BeH BTOPOMY.

44
..
KOHTpOJ/bH8f1 pa60Ta no TeM8M 9 2 H 93
-4
BapHaHT 8
I. On.HH H3 BHeWHHX yrJloB rpeyrOJ1bHHKa paBeH 124 0 . rpallycHble Me-
pbl yrJloB TpeyronbHHKa, He CMe>KHbIX C JlaHHblM BHellIHHM yrJIOM, OT-
HOCHTCH K3K 3 : 5. HaAD.H1"e rpaJlycHYIO Mepy 60JlbllierO H3 3THX yr JIOS.
2. B TpeyrOJ1bHHKe ABC yrOJI A paBeH 20o yrOJJ B paBeH 120 0 . npH-
Mble, con.ep>KalUHe BblCOTbl AAI Ii BB I , nepeceKalOTCS1 B TOlJKe H. Hai1-
.QHTe rpallYCHYIO Mepy yrJla AHB.
3. I-Ia pHCYHKe M306pa>KeHbl D.Be napbl napaJlJ1eJ1bHbIX np5lMblX: a, b H
c, d, a T8KiKe OTMell€Hbl yr.1lbl a H (3. HaHIlHTe rpan.YCHYIO Mepy YI'JI3 (3,
eCJ1H H3BeCTHo, liTO OH 60JlbWe yrJla Q pOBHO B lIBa pa3a.
b
a
4. nepHMeTp paBH06eJ1peHHOro rpeyrOJIbHHKa paBe1 18, a CYMM3 HeKO-
TOpblX llBYX ero CTOpOH pasHa 10. Haii,QHTe fl.J1HHY OCHOBaHHH 3Toro
rpeyronbHHKa.
5. lloK3>KHTe, lITO eCJIM 6HcceKTpHca H BbICOTa, npOBelleHHble K rHno-
TeHY3e OHoro npHMoyrobHoro TpeyronbHHKa paBHbl COOTBeTCTseH-
HO 6HcceKTpHce H BbICOTe, npOBelleHHblM K rHnOTeHY3e llpyroro npH-
u
Moyro.nbHoro TpeyrOJIbHHK3, TO nepBblH H3 3THX TpeyroJlbHHKOB paBeH
BTOPOMY.
BapHaHT 9*
1. O.QHH H3 BHeWHHX yrJIOB paBHo6e.QpeHHoro TpeyroJlbHHKa paBeH 150 0 .
HaAn.HTe rpallycI1YlO Mepy 60JIbWerO 113 BHyrpeHHHx yrJIOB 3Toro Tpe-
yroJlbHHKa.

KOHTpOJlbHaH pa60Ta no TeMaM 2 H 3
45
2. B TpeyroJlbHHKe ABC yroJl B paBeH 115°, AD 6HcceKTpHca yr.na
A, yroJl C MeHbwe yrJl3 ADB B ceMb pa3. HaHDJ1Te rpan.ycl-fYIO Mepy
yr JI3 C.
3. B napaJ1JIeJlOrpaMMe ABC D npOBeJleHbl 6HcceKTplicbl yrJIOB B M
C, nepeCeKalOll.lHe CTOPOHY AD B TOtlK3X M H L COOTBeTCTBeHHO.
113BeCTHO, liTO BC 15, AI L 7, a TOtlK3 nepeCeqeHHH npHMblx BJv!
I-I C L JIe)l(HT BHyrp" napaJlJleJ10rpaMM3, CM. 113HHbIH HH>Ke PHCYHOK.
Ila11.nHTe llJ1HHY CTOpOHbl AB.
A
1
L
7
M
-I. nepHMeTp paBHo6en.peHHoro TpeyrOJIbHHK8 paBeH 50, a pa3HocTb
11t'1«)TOPbIX llBYX era CTOpOH pasHa 11. HaHllHTeJIJlHHY OCHOBaHHH 3TO-
I. I 'l'f1eyrOJlbHHKa, eCJIH llOnOJ1HHTeJ1bl.JO H3BeCTHO, 4TO 3TOT Tpeyro.nb-
u
IIIII{ TynoyrOJ1bHbIH.
r t, .J lC)Ka)f{HTe, liTO ec.nH D.Be CTOpOHbl H 3aKJ1lO t leHHaS1 Me>K.n.y HHMH Me-
111.111;-) OJIHOrO rpeyrOJJbHHK3 paBHbl COOTBeTCTBeHHO HBYM CTopOHaM Ii
.. U U
I.Ih,III()lleHHOII Me)l(n.y HHMH MeIlHaHe ltpyroro TpeyrOJIbHHKa, TO nepBblH
U I . J'IIX rpeyrOJIbHHKOB paBeH BTOPOMY.
BapHaHT 1 0*
I .) 'IIII 113 BHeWHHX yr J10B pasHo6eJlpeHHOro TpeyrOJIbHHKa paBeH 112 0 .
It, III , III T e rpallycHYIO Mepy MeHbllJerO "3 BHYTpeHHHX yr JIOR 3Toro Tpe-
. I I . III ,11111<£1.
I I \ II )(\yrOJIbHHKe ABC yrOJI C paBeH 30°, AD - 6HcceKTpHca yrJI3
, \' II ),11 B 60JlbWe yr Jla ADB B TPM pa3a. Hai1JtHTe rpaJlycHYIO Mepy
· L' II.

46
.
KOHTpOJlbH8n pa60r8 no TeMaAf 2 H 3
3. B napaJIJleJlOrpaMMe ABCD npOBelleHbI 6HcceKTpHcbl yr.nOB B ".
C, nepeceKalOUlHe CTOPOHY AD B TOIIKax A.I H L COOTBeTCTBeHHO.
Vl3seCTHO, liTO BC 23!vI L 9, a "OlJKa IlepeCeLJeHH5J npHMblx B1\l
H C L JIeiKHT BHe napaJ1JIeJIOrpaMMa, CM. l!aHHb111 H11)Ke PHCYHOK. }-iaH-
llHTe JlJIHHY CTOpOHbl AB.
B
23
c
A
M
9
L
4. nepHMeTp paBH06eJlpeHHOI'o TpeyrOJJbHHKa paBeJI 70, a
HeKoTopblX llBYX era CTOpO1 paBla 16. l-IaHllHTe n..n.11HY OCHOBaHHSI ·
ra TpeyrOJIbl-lHKa, eCJ1J.-f JlOnOJlHHTeJ1bl-iO 113BeCTHO, tIT() ;rOl'  
u
HHK - TynoyronbHblH.
5. MellH3J-lbl AAI H BBI Tpeyr0J1bl1liKa ABC nepeCeK3I()TCS1 B T"OlIKe
lloKa>l(HTe, liTO AAI : AlA,! BAI : B 1 .1',! 2: 1.

9 4 . nop.o6ne cjJHIJ'P
 4. nOAo6He IJJHryp
47
4.1. 3aAaI.JH c npaKTHlJeCKHM COAep>KaHHeM
I. 4 To6bl npoeKTOp P nOJlHOCTblO OCBelllaJI 3KpaH ?__
:. 3 1S  ,"
A BblCOTOH 140 CM, ero Heo6xoJlHMO paCnOJlO>KHTb : ,,'::.-.r"--
I ,.. ..--
315 R ,,,.... ---A B
Iia paCCTOHHHH, He MeHbweM, lJeM CM 01 3Kpa- a.li::::----
Ita A. Ha KaKOM HaHMeHbweM paCCTOSlHHH (B caH- ':::jl-_:::
.....
'1'1,1 MeTpax ) OT 3KpaHa B BblCOTOH 220 CM HY>KHO ",
1)(lcnOJIOJKHTb npoeKTop, QTo6bl OH nOJlHOCTblO OCBem3J1 3TOT 3KpaH?
920CM
t' ,'1
I ? ,,' I
--
I ,,' ,........ I
I " --
, ",.-- .
I .,..- , - ..
D =---- A IB
cl:l. l =:---" -- I
......
... ... -- - J
' ........ -  
... ... ...
" --... I
.... ... ..
, .... 'I
"
... ... I
....
2. 4ro6bl npoeKTop P nOJ1HOCTblO OCBell.lan 3KpaH
J l IbICOTOH 230 CM, era Heo6xoJlHMO paCnOJIO>l<HTb
11:1 paCCTOHHHH, He MeHbweM, 4eM 920 CM OT 3KpaHa
fl. I--Ia K3KOM HaHMeHbweM paccToSlHHH (B C8HTH-
f\1( 'rpax) OT 3KpaH3 A BblCOTOM 130 CM HYIKHO pac-
lit )JIO)l{HTb npoeKTop, lJTo6bl OH nOJlHOCTblO ocselUaJl 3TOT 3KpaH?
:1. 4eJ10BeK pOCTOM 180 CM CTOHT
11.1 1)1CCTOHHHH 9 warOB OT cToJl6a c
'Iu 111;tpeM. npH 3TOM OH or6pacbIsa-
I I 1 t'llb JI.J1HHOH 6 warOB. OnpeJleJIHTe
IlitllMCPHYIO BblCOTY (B MeTpax) CTOJI-
'I" I' f\.1eCre c c1JoHapeM.
-I. (I eJlOBeK CTOHT H3 paccToSlHMIf
III 111;11'08 or cToJI6a C cpoHapeM. npM
'III' ()II oT6pacbIBaeT TeHb DJlHHOH 5
111.11 tilL OnpelleJIHTe npHMepHbIH pOCT
, '1"I'l'I{£l (8 MeTpax), eCJIH H3BeCTHO,
'.I'I'\I.ll'()Ta cToJI6a BMeCTe c cfJOHapeM
I t II I III '.,1 SleT 4,5 MeTpa.
?
.
180 CM
6 wares
9 waroB
4.5M

5 warns
10 warns

48
..,
4. nOJl.06Jfe cjJnryp
5. ,anSi OnpeHeJIeHHSI BblCOTbI !lepeBa MO>KHO HCnOJlb30BaTb 3epKaJlO
TaK, K3K nOK33aHO H3 JlaHHOM HH>Ke pHcYHKe. J1yq CBeTa CE, OTpa>Ka..
SlCb OT 3epK8JJ3 B T04Ke E, nOnaJlaeT B rJl33 4eJIOBeKa - TOlfKY A, np"
3TOM COrJlaCHO 3aKoHY OTpa>KeHHH (yroJl naeHHH paBeH yrJlY OTpa>Ke-
HHH) HMeeT MeCTO paBeHcTBo LCED LAEB. OnpeHeJIHTe BbICOTY
JlepeB3, ecnH AB 150 CM, BE 250 CM, DE 15 M. OTBer Bblpa-
3HTe B MeTpax.
B
3SpK8170
D
6. !l.nH OnpelleJIeHHH BbICOTbl 1!epeBa MO)KHO HCnOJlb30BaTb
KaK YKa3aHO B npeJlbl,QYl.l.leH 3aJlalJe. OnpelleJ1I1Te BblCOTY LlepcB3,
AB 170 CM, BE 300 CM, DE - 24 M. OTseT BbIpa3HTe B MeTpax.
7. Ha pHcYHKe H306pa)KeH KOJlO1I.eu
«>KypaBJ1b». KopoTKoe nJIetJQ HMeeT
J].JIHHY 1,3 MeTpa, a llJIHHHOe nJIeqo -
3,9 MeTpa. Ha CKOJIbKO MeTpoB ony-
CTHTCH BeJ1.pO, KOrlta KOHeLl KOpOTKO-
ro nJlelJ3 nOD.HHMeTCH Ha 1 MeTp?

8. Ha pHCYHKe H306pa>KeH KOJIOlleu
«)I{ypaBJIb». KopoTKoe nJlelJO "MeeT
,n.nHHY 0,7 Merpa, a ,llJIHHHOe nJleqo -
1,8 MeTpa. Ha CKOJIbKO caHTHMeTpoB
onycTHTcSI Be,npo, KOrJl8 KOHeLl KOPOT-
Koro nJleqa nO)tHHMeTCH Ha 42 CM?
--
9. Ha .n3HHOM HH)I{e pMcYHKe nOK33aHO, KaK MO>KHO Onp e lleJ1HTb
PHHY peKH, paccMaTpHB8H nOJto6Hbie TpeyrOJIbHHKH ABC H AB 1 Cle

94. nOo6HeHp 49
HaHJlHTe lllHPHHY peKM (B MeTpax), eCJ1H H3BeCTHO, liTO AC 70 M,
ACt 25 M, ABI 30 M.
B
----
---
---
----
---._
---:.B
- - - I
4.2. 3aAaqH Ha Bbll.JHCJleHHe
1 O. Ha cTopoHe AD napaJIJIeJlOrpaMMa ABCD B3HTa T01.fKa K, P-
1"()(IKa nepeceLleHHH npHMbIx BI( HAC. HaHJlHTe OTHoweHHe AP : CP,
('l-JIH H3BecTHo, qTO: a) AK: DK 4: 5; 6) AK: DK k.
11. Ha llHarOHaJ1H AC napaJIJleJlOrpaMM3 ABCD 83HTa TOqKa P,
/\. - TOLJK3 nepece4eHMSI npHMbIx BP HAD. HaHllHTe OTHorneHHe
,/{ : DI(, eCJ1H H3BeCTHO, LITO: a) AP: C P 3 : 5; 6) AP: CP 7: 3;
I\)AP:CP p,npH4eMp<1; r)AP:CP p,npHlJeMp> 1.
12. Ha !tHarOH8J1H AC napaJlJIeJlOrpaMMa ABCD B3HTa TOtIKa P,
t ,J T04Ka nepeCeqeHHH npSIMblx BP H CD. HaHJlHTe OTHorneHHe
f 'tJ: DQ, eCJIH H3BeCTHO, LITO: a) AP: CP 2 : 7; 6) AP: CP 8: 5;
.,).\P:CP p, npHlJeM p < 1; r) AP:CP p, npHlJeM p > 1.
I :1. Iia llHarOHaJiH AC napaJlJIeJlOrpaMMa ABCD B3HT3 TOqK3 P,
1\ r04Ka nepece4eHHH npHMblx BP H AD. HaH,LlHTe OTHOllleHHe
"': (;P, eCJIH M3BeCTHO, qTO: a) AD: DK 5: 6; 6) AD :DK 3: 2;
..) :\'): DI( k, npHlJeM k  1; r) AD: DK k, npHlJeM k > 1.
1.1. Ilcl llHarOHaJIH AC napaJIJIeJlOrpaMMa ABCD B3HTa TOIlKa P,
t -' I ()IIKa nepecelleHHSI npHMbIx BP M CD. HaHllHTe OTHOIlleHHe
, I' (' P, eCJIH H3BeCTHO, qTO: a) CD : DQ 3 : 4; 6) CD: DQ 5 : 3;
'I , 'I': DQ k, npHlieM k  1; r) CD :DQ k, npHqeM k > 1.

50
..
94. nOJI06He CPHryp
15. B TpeyroJlbHHK ABC BnHC3H KBa.npaT K LM N TaK, liTO BepWHHbl
K 11 L JIe>KaT Ha CTOpOHaX AB H AC COOTBeTCTBeHHO, a BepWHHbl M
H N H3 CTopOHe BC. HaHJ!HTe J]JIHHY CTOpOHbl 3Toro KBaJJ.paTa, eCJ1H
BC a, AH h, rlte AH BblCOTa TpeyroJlbHHKa ABC.
16. B TpeyrOJIbHI1K ABC BnHcaH pOM6 CDEF TaK, llTO BepWHHbI D,
E H F ne>KaT Ha CTopOHax AC, AB H BC COOTBeTCTBeHHO. I-laH!lHTe
DJIHHY CTOpOHbI 3Toro pOM6a, eCJlH BC a, AC b.
17.* Ha 60KOBbiX cTopoHax AB H CD rpaneUHH ABC D B3HTbI TOll-
KH M H N TaK, 4TO AM : BN! DN : CN p: q. HaHllHTe ,llJIHHY
oTpe3Ka ]VIN, eCJ1H H3BeCTHO, 4TO AD a, BC b.
18.* Ha 60KOBbIX cTopoHax AB H CD TpaneUHH ABCD B351Tbl TOlJ-
KH M H N TaK, liTO orpe30K MN napaJ1.neJIeH OCHOBaHH5IM TpaneUHH.
HaHD-HTe OTHoweHHe AM: BM, eCJlH AD a, BC b, MN l.
19.* Ha CTopOHax AC H AB TpeyroJlbHHKa ABC B3HTbi TOQKH B 1 H
C 1 , oTpe3KH BBI Ii CC l nepeceKalOTC5I B TOlJKe P. HaH)]J1Te OTHorne-
HHeBP: PBI, eCJlH H3BeCTHO, lJTO: a) AB 1 : BIG 4: 5, AC l : CIB
3 : 2; 6) AB 1 : B 1 C p, A G 1 : C 1 B q.
20.* Ha cTopoHax AC H AB TpeyrOJIbHHKa ABC B3HTbi TOlJKH Bl Ii
C 1 ,oTpe3KH BBl H CC l nepeceKalOTcH B T04Ke P. HaHJtHTe OTHorne-
HHe AC 1 : ClB, eCJIH H3BeCTHO, 4TO: a) BP: PB I 4: 3, CP: PCI
9:4; 6)BP:PB 1 p, CP:PC 1 q.
4.3. 3aAallH Ha AOKa3aTeJlbCTBO
21. Ha .QHarOHaJIH AC napaJlJIeJIOrpaMM3 ABC D B3HT3 TOqK3 K.
npSlMaH BK nepeceKaeT np5lMbie CD H AD B TOqKaX L H M COOT-
BerCTBeHHO. JloKa>KHTe, qTO BK 2 KL. KM.
22. OTpe30K G H BblCOTa npHMoyrOJIbHOrO TpeyrOJIbHHK3 ABC c
rHnoTeHY30H AB. JloK3>KHTe, qro: a) CH2 AH .BH;
6) AC 2 AH -ABe

4. nOP.0611e cjJHryp 51
23. OTpe30K AH BblCOTa TpeyrOJ1bHHKa ABC. 113 TOlJKH H npo-
BelleHbl nepneHJlHKYJ1SIpbl H P 11 HQ K CTOpOHaM AB H AC COOT-
BeTCTBeHHO.lloKa)l{HTe, 1..JTO lpeyrOJ]bHHK APQ nOllo6eH TpeyrOJ1bl.iH-
KY ABC.
24. Ha CTOpOl1bl BC H CD napa.nJIeJlOrpaMMa ABCD (11J1H H8 I1X
IlpO!lOJ1}f{elliH) oIlylltelbl nepneI-IJJJiKYJ151pbl A!\.f Ii AN. llOKa>KHTe, lJTO
l"peyrOJlbHHK lv/AN n0l106eH TpeyroJlbHHKY ABC.
25. OTpe3Kli BBl H CC I - BblCOTbl TpeyroJlbHHK3 ABC, B KOTOpOM
Illf OltliH 113 yrJloB I-Ie HB.nHeTCH npSlMbiM. llOK3)1{HTe, lJTO TpeyrOJIhHI1K
.' \ B 1 C} n0J106el-f rpeYI'OJIbHHKY ABC.
26. B paBHo6ellpeHHoM TpeyrOJIbllliKe ABC C OCHOB3HHeM AB IlpO-
Itl\JlCH3 Bblcora AH. llOKa)KHTe, liTO AB 2 2 BC · BH.
27.* llOKa)KH1e, lrro Ii Jllo60ii rpanelll1H flpHM851, 11POXO,UJJU1351 4epe3
u u
I ()IIKY nepece4eltHH Jllal"OHaJ]eH H T0111<Y nepeCeqCHHH rlpOllOJl>Kel-lHH
I H)I<OBbIX CTOpOH, JleJII1T o6a OCI-IOB3HHH 3TOH rpaneLlI1H nOnOJ1aM.
:H.:i: TOIIK3 P Jle)KH1 Ha MeJllaHe C M TpeyrOJlbl-lHKa ABC. OpS11\1ble
, I) JI BP nepeceKalOT CTOpOHbl TpeyroJlbHHKa B T04Kax Ai Ii B 1 COOT-
I'C'll'"rBeI-IIJo.1l0K3)1(IITe, liTO npSlMa51 AIB] napaJ1J1eJlbHa np5IMOH AB.
:)!):-=-. OTpe30K C L - 6HcceKTplica TpeyrOJJbHHKa ABC. ,UoK8MHTe,
'III) ,1 L : B LAC : B C.
:Ift.\: npHM3H l nepeceKaeT CTOpOHbl AB H AD napaJlJleJIOrpaMMa
, III f D B TOIIKax E H F COOTBeTCTBeHHO, G - T04 K3 nep e ce4e HHH

52
95. TeopeM8 [7JfrjJaropa. CHHYC, KOCJfHYC, r8HreH('
.
 5. TeopeMa "Haropa. CHHYC, KOCHHYC, TaHreHC
5.1. TeopeMa "H4Jaropa
1. BblcoTa, npOBeJleHHaH K OCHOBaHHIO paBH06eJl.peHHOro TpeyroJlbHI1-
Ka, paBH8 12, a OCHOBaHHe pasHo 10. HaH.QHTe llJlHHY 60KOBOH CTOpO-
HbI 3Toro TpeyroJlbHHK3.
2. BblcoTa, npoBelleHHaH K OCHOBaHHIO paBHo6en.peHHoro TpeyrOJlbHH-
Ka, paBI1a 40, a OCHOS8HHe pasHo 18. 1--laI!lI1Te LLJIHHY 60KOBOH CTOpO-
Hbl 3Toro TpeyroJlbHHKa.
3. DOKOB3S1 CTopOHa paaHo6ellpeHHoro rpeyrO.flbHHK3 paSHa 17, a OC-
HosaHHe paBI10 16. HaHllHTe JlJIHHY BbICOTbl, npOBeJleHHOH K OCHOBa-
HHIO 3Toro TpeyrOJ1bliHKa.
4. DOKOB3H CTopOHa paBHo6eJtpel-liiOrO rpeyrOJIbHHK3 paBH3 35, a Bbl-
COT3, npoBeAeHHaH K OCHOBaHHIO, pasHa 21. l-Iai1llHTe LLJIHHY OCHOBa-
HHH 3Toro TpeyroJlbHHKa.
5. l{aTeTbl npHMoyroJlbHoro rpeyrOJIbHHK3 paBHbI 20 H 21. 11aiiltHTe
.. "
LLJIHHY Me.QHaHbI, npoBe,QeHlioH K rHnoTeHY3e 3Toro TpeyrOJ1bHHKa.
6. OllHH H3 KaTeTOB npHMoyrOJ1bHOrO TpeyrOJIbHHK3 paBeH 24, a JlJ1HHa
MellHaHbl, npOBeJleHHOH K rHnOTeHY3e, pasHa 13. HaHAHTe ,UJIHHY BTO-
pora K3TeT3 3Toro TpeyroJlbHHKa.
7. llHarOHaJlH pOM6a paBHbl 14 H 48. HaHJlHTe CTOPOHY 3Toro pOM6a.
8. llHarOH3J1H pOM6a paBHbI 10 H 24. HaHllHTe nepHMeTp 3Toro pOM6a.
9. nepHMeTp pOM6a paBeH 148, a pa3HocTb llJlMH ero J!HarOH3JIeH paB-
H8 46. l-IaH,QHTe .LlJIHHY MeHbweH H3 AHarOH3J1eH 3Toro pOM6a.
1 o. nepHMeTp pOM6a paseH 164, a pa3HocTb ,UJIHH era !lHarOH3J1eH
paSHa 62. H3HltHTe JlJIHHY 60JIbweet 113 J!HarOH8J1eH 3Toro pOM6a.

95. TeopeM8 nHcjJaropa. CHHYC, KOCHHYC, r8HreHC
53
11. I-Ia pHcYHKe H306pa)l{eH3 rpaneUHSI ABCD, lL11H KOTOpOH H3BeCT-
Hbl JlJ1HHbI CTOpOH AB, BC CD H oTpe3Ka AH, fJIe H OCHOBaHHe
BblCOTbl BH. l-IaHllHTe J1JIHHY CTOpOHbl AD.
B 23 C
13
5
A H
D
12. Ha pHcYHKe H306pa)KeHa TpaneLlHs:I ABG D, 1lJ1S1 KOTOpOH H3BeCT-
IlblllJIHHbl CTOpOH AB, CD H Orpe3i<OB AH, DH, r-ne H OCHOB3HHe
lblCOTbl BH. HaHllHTe D.JIHHY CTOpOHbl BG.
B
20 37
H
A
D
13. Ha pHCYHKe H306pa>KeHa TpaneUHH ABCD, LtnH KOTOpOH 113BeCT-
111)1 JlJIHlibI CTOpOH BC, CD H oTpe3KoB AH, DH, r,ae H OCHOBaHHe
"I.ICOTbI BH. HaHllHTe D.JlHHY CTOpOHbI AB.
B 25 C
70
70
A
H
D
IiI. JIecTHHua COeJlHHSIer TOllKH A H B.
I \I.U-()Ta K3)KH.OH cryneHH paSHa 16 CM, a
"11111:1 - 30 CM. PaCCTOHHHe MeiKUY TOqKa-
'.'11 ...1 H B COCTaBJI5IeT 34 M. HaHD-HTe BbI-
I 'II ', Iia KOTOpylO nOllHHMaeTCH J1eCTHHlla. A
· 'IIH'I' Bblpa3HTe B Merpax.
e
e
It
34M
If., .1lecTHHua CoeJlHHHer TOtlKH A H B, CM. PHCYHOK K npe.llbIJlyw.eH
..1 1 1.1 1 1('. Bblcora Ka)l(llOH c1)'neHH paBHa 14 CM, a JI.JIHHa 48 CM. Pac-
'11\llllll\ Me)f(ny TOllKaMH A Ii B COCT3BJlHeT 35 M. HaHJ1HTe BbICOTY, H8
... .1. tl )VI() no-aHHMaeTCSI J1eCTHHua. OTBeT BbIpa3HTe B MeTpaXe

54
.
5. leopeMa nHcjJarOpa. CHHYC, KOCIIHYC, raHreHC
16. Ha BepwHHax D.BYX eJIe" CHllHT llBe COpOKH. BbICOTa nepBoH eJIH
pasHa 14 M, BbICOT3 BTOpOH - 30 M, a paCCT05lHHe Me>KY eJ151MH pasHo
88 M. Ha KaKOM paCCT05IHHH BE OT flepBoH e&l1H HY}I{HO n0J10>KI1Tb Cblp
ll.TIH 3THX COpOK, lJTo6bl OHH H3XOllHJlliCb B paBHblx YCJ10BHHX, T.e. 4TO-
6bI paCCT05l1"le OT Hi'IX !to cblpa 6bIJ10 01lHHaKoBbIM? ()TBel' Bblpa3HTe B
MeTpax.
30M
14 M
.....
....
,
,
,
,


,
,
,
,
, ,
" E '
, ,
..-..:
.......
"
,
....
B
D
88M
17. Ha BepwHHax !tBYx eJ1eH CHllSIT llBe COpOKH. BblCOT3 nepBoft eJ1H
paBH3 20 M, BblCOTa BTOpOH - 24 M, a paccToHHHe MeIK1JY eJ1HMH pasHo
22 M. Ha K3KOM paccToSlHHH DE OT BTOpOH eJIJ1 HY/KHO 110JIO)l{I1Tb Cblp
JlJ1H 3THX COpOK, 4To6bl OHH HaXOJlHJlHCb B paBHbIX YCJlOBH51X, T.e. lITO-
6bl paCCTOHHHe OT HI1X D.O Cblpa 6blJ10 OIlHH3KOBbIM? OTBeT Bblpa311Te B
MeTpax.
20M
,
,
,
,
,
,
,
, '
, c'
'c'
,
,
,
,
24M
B
D
22M
18. Cre6eJ1b K3Mbiwa BblcrynaeT 113 BOJlbl H3
0,8 M. Era nepxHHH KOHeu OTKJlOHliJIH OT sepTH-
KaJIbHOrO nOJlO)KeI1H5I H3 1,2 M, H OH OKa38J1CSI H3
ypoBHe BOJlbl. HaMJlHTe rJIy6HHY o3epa B TOM Me-
CTe, rn-e paCTeT KaMbIW. OTBeT BbIpa3HTe B MeTpax.
:E
00
....
o
1 2 1

5. TeOpeMa nHcjJarOpa. CHHYC, KOCHHYC, T8HreHC
19. CTe6eJ1b KaMblwa BbIcTynaeT H3 BO,UbI H3
0, 7 M. Era BepxHHH KOHeu OTKJIOHHJIH OT BepTH-
KaJIbHOrO nOJIO)l{eHH5I Ha 1,4 M, Ii OH OK33aJlCH H3
ypoBHe BOllbl. HaHJ1HTe r J1y6HHY o3epa B TOM Me-
CTe, r-ae pacrer K3MbIW. OTBeT BbIpa3HTe B MeTpax.
55
2
r--
...
o
14M
20. nepHMeTp npHMoyroJlbHHK3 paBeH 38, a ,QHarOHaJlb paBHa 11.
1-laAAHTe nJI01ll8J!b 3Toro np5lMoyrOJIbHHK3.
21. nepHMeTp npHMoyroJlbHHK3 paBeH 82, a AHarOHaJIb paBHa 29.
IlaHAHTe JLTII1HY 60JIbweti H3 CTOpOH 3Toro npSlMoyrOJIbHHKa.
22. nepHMeTp npHMoyroJlbHHKa paBeH 124, a llHarOHaJIb pasHa 50.
IlaIJlHTe JI.J1I1HY MeHbweti H3 CTOpOH 3Toro npSlMoyroJlbHHK3.
2:J. B npSlMoyrOJ1bHOM TpeyrOJIbHHKe MHHa MeJ1HaHbl, npoBelleHHoti K
1IIIIOTeHY3e, paBH3 25, a 1lJIHH8 BblCOTbl K rHnoTeHY3e paBHa 7. HaHJlH-
1(' JlJJHHY 60JIblllerO H3 KaTeTOB 3Toro TpeyroJlbHHKa.
'1. B npHMoyrOJIbHOM TpeyrOJ1bHHKe MHHa MeJlH3Hbl, npoBeneHHoM K
I 1111()TeHY3e, pasHa 17, a ,llJ1HHa BbICOTbl K rHnoTeHY3e paBHa 15. HaH-
1111'(,' 1lJIHHY MeHbwero 113 KaTerOB 3Toro TpeyronbHHK3.
:)!). 13 npHMoyrOJIbHOM TpeyrOJIbHHKe ABC c npHMbIM yrJIOM C npo-
III'Ilt'lla BbIcora CH. 113BecTHo, QTO AH 20, BH 22,05. HaHJ1M-
III 1I.111'iHY KareTa AC.
:'fi. 13 npSIMoyroJlbHoM rpeyroJlbHHKe ABC c npHMblM yrJloM C npo-
,.. '1"11£1 BbICOT3 CH. 113BecTHo, QTO CH 3, AH 2,25. HaHJlHTe
I 'II lilY KareT3 BC.
.'/  1\ rlpHMoyrOJIbHOM TpeyrOJIbHHKe ABC c npHMblM yrJloM C npo-
... 1.'11;1 nblcora CH. H3BeCTHo, liTO CH 12, AB 25. HaHJlHTe
I 'II lilY f)()JIbWerO H3 K3TeTOB 3Toro TpeyrOJIbHHK3.
'" 1\ IlpHMoyrOJIbHOM TpeyroJlbHHKe ABC c npHMblM yrJIOM C "PO-
I,' It III t III)ICOTa C H. 113BecTHo, lJTO C H 24, AB 67,6. HaHJI.HTe
, "'IIV 1('llblllero 113 K3TeTOB 3Toro TpeyroJlbHHKa.

56
5. TeOpeM8 nlfcparOpa. CJIHYC, KOCHHYC, TaHreHC
29.* [{J1HHbl CTOpOH TpeyroJlbHHKa paBHbl 5, 6, 9. HaHJlHTe llJIHHY
.. U U
BbICOTbl, npoBe.QeHHoH K MeHbWeH 113 CTOpOH 3TOrO TpeyrOJIbHHKa.
30.* [{JIHHbl CTOpOH TpeyrOJIbHHKa paBHbI 2, 10, 415. HaHltHTe II.JlH-
HY BbICOTbI, npoBelleHHoH K 60Jlbweti H3 CTOpOH 3Toro rpeyrOJ1bHHKa.
31.* llOKa)KHTe, 410 B JlI060M napaJIJIeJlOrpaMMe CYMMa KB3J1paTOB
u
l!BYX era l!HarOHaJlel1 paBHa CYMMe KBallpaTOB Bcex ero c.ropOH.
32.* llOKa>KHTe, liTO eCJlH a, b, C JI.nHHbl CTOpOH TpeyrOJ1bHHKa, TO
.. u
JI)lHHa me MeJlH8HbI 3TOrO TpeyrOJIb H H K 3, npOBeJleHHOH K CTopOHe lL11H-
5.2. CH"YC, KOCHHYC, TaHreHC
33. HaHltHTe TaHreHC yrJIa C TpeyrOJIb-
HHK3 ABC, H306pa>KeHHOrO Ha pHcYHKe.
B
c
A
34. HaHllHTe CHHYC yrJla A TpeyrOJIbHH-
Ka ABC, H306pa)KeHHOrO Ha pHcYHKe.
B
A
c
35. HaHJlHTe TaHreHC yr Jla B TpeyrOJIb-
HHKa ABC, H306pa>KeHHOrO Ha pHCYHKe.
A
36. HaHl{HTe KOCHHYC yr JIa A Tpeyronb-
HHK3 ABC, H306pa>KeHHOrO Ha pHCYHKe.
c
A

5. TeOpeltla flJfcjJaropa. CIIHYC, KOCHHYC, raHreHC
57
37. HaHllHTe T3HreHC yrna B TpeyroJlb-
1-I"tKa ABC, It306pa)KeHHOrO Ha pHCYHKe.
c
A
38. I-IaH!lHTe TaHreHC yrJla A TpeyrOJIb-
1IIIKa ABC, H306pa)KeHHOrO l-fa pHcYHKe.
A
B
c
:J9. HaHJlHTe TaHreHC yrJ13 B TpeyroJlb-
11I11<a ABC, H306pa}KeHHOrO Ha pHCYHKe.
B
,I(). (;aHD.HTe T3HreHC yrJIa A TpeyrOJIb-
11111\;) ABC, H306pa/KeHHOrO Ha pHcYHKe.
c
B
A
.1 I. 13 npHMoYI'OJlbHOM TpeyrOJIbHHKe ABC c npHMbIM yrJloM C 113-
'" 'f I rlll)lllJJIHbl CTOpOH: BC 12, AB 13. HaMJlHTe tg LA.
,I:., I npHMoyroJlbHoM TpeyroJlbHHKe ABC c npHMblM yrJloM C H3-
.11 I 1111,1 JlJIHHbl CTOpOH: BC 4, AC SV6. HatiJlHTe sin LA.
1.1. 1\ l)aBH06ellpeHHoM TpeyroJlbHHKe ABC C OCHOBaHHeM AC 60KO-
h '" t' I ()Jl0lla AB pasHa 32, a cos LA 0, 7. HaHJ!HTe J1JIHHY AC.
II I  l)cIBII06eJlpeHHOM TpeyrOJIbHMKe ABC C OCHOB3HHeM AC 60KO-
,, · It)( )()I-Ja AB pasHa 15, a sin LA O,4V6. HaHJlI1Te J1JIHHY AC.
I. I  1)(lBli06eJlpeHHOM TpeyrOJIbHHKe ABC 60KOBaSi cTopoHa AB
Hili I Iii', OCHOB3HHe AC P3BHO 16. HawAHTe tg LA.

58
5. TeOpeMa nHcjJarOpa. CHHYC, KOCHHYC, raHreHC
46. B paBHo6ellpeHHoM TpeyrOJIbHHKe ABC C OCHOB3HHeM AC 20
BbICOTa, npoBen-eHH3S1 K OCHOBaHHIO, pasHa 5 21. HaHJlHTe cos LA.
47. B paBH06eJlpeHHOM TpeyrOJIbHHKe ABC C OCHOB3HHeM AC BbICQ-
Ta, npoBe,neHHaSi K OCHOBaHHIO, pasHa 12, a sin LA 0,4/5. HaHllMTe
JlJIHHY AC.
48. B paBHo6en.peHHoM TpeyroJlbHHKe ABC C OCHOB3HHeM AC BbICO-
ra, npOBeneHH3S1 K OCHOB3HHIO, pasHa 14, a cos LA 0,1 V2. HaH,QHTe
JlJIHHY AC.
49. B npHMoyrOJIbHOM TpeyrOJIbHHKe ABC c np5lMblM yr.nOM C H3-
r:
BeCTHbl rHnOTeHY3a H TaHreHC yrJ1a: AB 39, tg LA IV 2 .
DJIHHY KaTeT3 AC.
50. B npS1Moyro.nbHOM TpeyrOJ1bHHKe A BC c npHMbl M yrJ1 0M C -
DJIMHY KaTeTa BC.
51. B paBH06eJlpeHHOM TpeyrOJIbHHKe ABC C OCHosaHHeM AB
yroJl np" BepwHHe C paBeH 120 0 . HaH,LIJ1Te D.JIHHY BbICOTbI AH.
52. B paBHo6eJtpeHHoM TpeyroJlbHHKe ABC C OCHOBaHHeM AB H3-
BeCTHbI 60KOBaH CTOpOH3 H yroJl np" OCHOBaHHH: BC 7, LB 15 0 .
HaHJlHTe DJIMHY BbICOTbI BH.
53. B paBHo6e,npeHHoM TpeyrOJIbHHKe ABC C OCHOBaHHeM AC
BaSI CTOpOH3 paBHa 11, a yroJl npH BepwHHe B paBeH 120 0 .
AH BbIcor3 3Toro TpeyroJlbHHK3. HaHHTe JI..nHHY oTpe3K3 CR.
54. B paBHo6e,npeHHoM TpeyrOJIbHHKe ABC C OCHOB3HHeM AC .
BaSI CTopOHa paBHa 20, a cos LC 0, 7. OTpe30K AH BblCOTa
TpeyrOJIbHHKa. HaHD-HTe DJIHHY oTpe3Ka CH.
55. B paBHo6e.npeHHoM TpeyroJlbHHKe ABC C OCHOBaHHeM AC
BaSI CTopOHa paBHa 14, a cos LC 0,6. OTpe30K AH - BblCOTa
TpeyroJlbHHKa. HaHJlHTe D.JIHHY oTpe3Ka BH.

9 5 . TeopeMa nlfcjJaropa. CIIHYC, KOCHHYC, r8HreHC
59
56. B npHMoyrOJIbHOM TpeyrOJ1bHHKe ABC c npSlMblM yrJloM C npo-
BelleHa BblCOTa C H. 113BecTHo. 4TO AC 4, AH 15. HaHJ{HTe
(OS LB.
57. B npSI1oyroJ]bHoM TpeyroJlbHHKe ABC c npHMblM yrJ10M C npo-
lellel-fa BblCOT3 C H. 113BecTHo, liTO AC 20, C H 5V7. HaI111HTe
:-;ill LB.
58. B flpHMoyrOJ1bHOM TpeyrOJ1bHHKe ABC c npHMblM yrJloM C npo-
l\eJleHa BblCOTa C H. 113BecTHo, lITO cos LA 0,2, AC 8. I-IaHlll.ITe
JlJIHI1Y oTpe3K3 AH.
59. B npHMoyrOJ1bHOM Tpeyro.l1bHl1Ke ABC c npHMbIM yr.nOM C npo-
'\l'JICI-Ia Bblcora C H. 113BecTHo, 4TO sin LA 0,3, AB 14. HaHJlHTe
'1.rllfHY oTpe3Ka BH.
(j(). B np5lMoyrOJ1bHOM TpeyrOJIbHHKe ABC c npHMblM yrJ10M C npo-
I\C'.'lCHa BblCOTa CH. H3seCTHO, liTO AC 15, CH 12. Hai1lIJ1Te
'I.IIIIIIY KaTeTa BC.
f) I. B npHMoyrOJ1bHOM TpeyrOJIbHHKe AB C C fJpHMb IM yr.noM C npo-
'I rllillY orpe3Ka BH.
f':'. IIYCTb a, b, c - J]JIHHbl CTOpOH npOH3BOJlbHOrO TpeyrOJIbHHKa, npo-
1III\t t.Ill'}J(alllliX BepWHl-laM A, B, C COOTBeTCTBeHHO. llOK3)K"ITe, lITO
-
SIll SIll SIll
t 'I" YlJJep>Kllel1He HOCHT Ha3BaHHe meopeMa CUHYCOB.)
f', I '. II YCTb a, b, C llJIHHbI CTOpOH npOH3Bo.Jlbl10ro TpeyrOJlbHliKa, r,Qe
'4 tl at .11;1 J!JlHHbl c npOTHBOne)KHT BepWHHe C. lloK3)KHTe, lITO cnpa-
,., . 'UIIU) CJIeJlYlOlllee paBelCTBO: c 2 a 2 + b 2 - 2ab · cos LC. (3TO
".. 1 1 ..-1\ 'lCI.llie I-IOCI--IT Ha3BaHHe meope.Ata KOCUflYCOB HJlH o606U{efl-
· , "" '(.JpeMa n uc/1ac.opa.)

60
KOHTpO.flbHan pa60ra no reMaM 4 H 95
KOHTpOJ1bHaSi pa60Ta no TeMaM 4 H 5
PeKOMeHn.yeMoe BpeMSI MH BblnOJIHeHHfI pa60Tbi 40 MHHyr.
PeKoMeHn.YMa51 LlIKaJIa OLleHHBaHHS1 pe3y.nbT3ToB:
..... 0-1 2-3 4 5
'-flllcno BepHblX peweHIII
WKonbHa 04eHKa 2 3 4 5
BapHaHT 1
I. 4T06bl npoeKTOp P no.nHOCTblO OCBellla.n 3KpaH ?,
:. 270CM , ... ,,:
A BblCOTOH 150 CM, ero Heo6xOLll1MO paCn0J10>KHTb : ".,:.,
Ha paCCTOHHHH, He MeHbweM, LJeM 270 CM or 3Kpa- pd.,f:::::.4= B
.:-==----
H3 A. Ha KaKOM HaHMeHbweM paccToHHHH (B caH- ""'::JL,--,
,
THMeTpax) or 3KpaHa B BbICOTOH 250 CM HY)I(HO ",
paCnOJIO>KHTb npoeKTop, 4T06bl OH nOJIHOCTblO OCBeLU3J1 3TOT 3KpaH?
2. BblcoTa, npOBeJleHHaH K OCHOBaHHIO paBHo6en.peHHoro
Ka, paBlia 48, a OCHOBaHHe paBHo 40. f1atiD.HTe l{JlHHY 60KOBOH
Hbl 3Toro TpeyroJlbHHK3.
3. nepHMerp pOM6a paBeH 100, a pa3HocTb llJIHH era JlHarOH3JIeH
10. HaHllHTe JlJIHHY MeHbweH 113 !lHarOHaJIeH 310ro pOM6a.
4. OCHoBaHHSI paBHo6e.npeHHo H T paneLlHI1 paBHbI 8 H 50, a
OCTporo yrJIa TpanellHH paBeH
u
3TOH TpaneLlHH.
5. B np5lMoyrOJIbHOM Tpeyr OJI bHHKe ABC c np HMbIM yrJIOM C
Ta BC.

KOHTpO.RbH8n pa60ra no TeMaM 4 H 5
61
BapHaHT 2
1. 4T06bl npoeKTop P nOJIHOCTblO OCBeLUan 3KpaH
B BblCOTOH 280 CM, ero Heo6xo,QHMO paCnOJIO>KHTb
l-fa paCCTOHHMH, He MeHbweM, lleM 840 CM OT Hero.
I-Ia KaKOM HaHMeHbweM paCCTOHHHH (B caHTHMer-
1)3X) OT 3KpaH3 A BblCOTOH 210 CM "ymHo pacno-
JIOiKHTb npoeKTop, 1.JTo6bI OH nOJlHOCTblO OCBe1ll3J1 3TOT 3KpaH?
840CM
,. ? ,.... .,
I . ,'1
I '.. .,. .. I
" --
I "'" I
I " '.... fill' ... _  .. .. ..
n "--=--::---- A IB
rl:l.l===---- ----1
«:.........-........
..., '-- -- J
... --... .. ..
...... ...  I
.... .. ...
" "'I
'..., I
'.
2. BblcoTa, npOBeJleHHaH K OCHOBaHHIO pasHo6e.apeHHoro rpeyroJlbHH-
I<a, pasHa 28, a OCHOBaHHe paBHo 42. HaH,QHTe llJ1HHY 60KOBOH CTOpO-
II!)I 3Toro TpeyrOJIbHHKa.
:J. nepHMeTp pOM6a paBeH 156, a pa3HocTb JtJ1HH ero D.HarOH3J1eH pasHa
.12. I-IaHJlHTe JIJIHHY 60J1bweti "3 !ll1arOHaJleH 3Toro pOM6a.
.1. [joJlbwee OCHOBaHHe paBHo6ellpeHHoH TpaneUHH pa BHo 70, 60KO-
I;ISI CTopOHa pasHa 14, a TaHreHC OCTporo yrJla paBeH
1{'llbwee OCHOB3HHe TpaneUMH.
!.. I rlpHMoyrOJIbHOM TpeyrOJIbHHKe ABC c npHMblM yrJIOM C 1'13BeCT-
'11.1 l<aTer H CHHYC yr .na: AC 7, sin LA 0,96. Hai1AHTe llJIHHY KaTe-
1.1 flC.
BapHaHT 3
I I Il\J10BeK pOCTOM 150 CM CTOHT H8
1',II'("r()S1HHH 7 warOB OT cToJ16a c
'1'..II;'1)CM. npH 3TOM OH or6paCblBa-
· I It 'III) ltJIHHOH 5 rnarOB. OnpelleJIHTe
1'1 tli t\1l 'f)HYIO BbICOTY (B MeTpax) CTOJl-
14 . 1'1l'lTe c <jJOHapeM.
?
.
150 CM
5 waros
7waroB
· ,,. '1\()I.Jafl CTOPOH3 paBHo6epeHHoro TpeyrOJIbHHKa paSHa 74, a oc-

62
.
KOH11)O.RbH8f1/Ja6or8 no Te\laAI 4 If '\,5
HOBal-llie paBHo 48. I iaflllliTC llJ1HHY BbICOTbl, 11poBelleHHoH K OCHOBa-
I-fHIO 3Toro TpeyroJlbHHK3.
3. nepllMerp Ilp5JMoyrOJ"lbl-fJ.-IKa paReH 126, a JlI1,JrOI-13Jlb paBH3 45. I-Iai1-
J111Te n.fl11HY 60.TIb11Ie11 113 CTOpOI-t 3101"0 flpHMoyro.nbHHKa.
4. B paBI106cJtpCHHOM lpeyr()JlbHlfKe ABC C OCH()13aHl-teM AC 60K()-
BaH cTopoHa paBIa 15, a yroJl npli Bepllllil-le B paBel-f 120 0 . Orpe30K
AH - Bl)ICOTa 3Toro TpeyrOJIhHlfKa. l-IaI1,QHTe Jl.nJ1HY ()Tr)e3Ka CR.
5. B npHMoyr'oT1blOM Tpeyro.Jlbl-IHKe ABC c rlp5l M hi M YI',nOM C li3BeCT-
KaTeTa AC.
Bapl1aHT 4
1. 4e..r1oBeK CTOHT H3 paccToHHI1H 8
waros (}1 CTOJI6a C cpOHapeM. n pli
3TOM OH oT6paCblBaeT TeHb D.JIHIOH 4
rnara. OnpelleJIHTe npHMepHblH pOCT
tle.TIOBeK8 (B MeTpax), eC.J1H H3BeCTHO,
liTO BblCOTa cToJI6a BMeCTe C cpOHapeM
COCTaBJlHeT 4,8 MeTpa.
4.8M
4 wara
8 warOB
2. DOKoBaSl CTopOHa paBHo6ellpel-lHOrO Tpeyro.nbl-lHKa paBH3 26, a BI)I-
COTa, npOBeJleHHa51 K OCHOBaHHIO, pasHa 10. HaHllHTe 1l"r]HHY OCHOBa-
HH5I 3Toro TpeyroJlbHHKa.
3. nepl1MeTp npHMoyrOJlbHI1Ka paBet-1196, a JlHarOIJa.TJI paSHa 82. (.Iaif-
u
,QHTe llJIHHY MeHbweH H3 CTOpOII 3Toro np5lMoyro.nL)11I1K3.
4. B paBHo6eJtpeHHoM TpeyrOJ1bHHKe ABC C OCHOBaHHeM AC H3BeCl'
Hbl 60KOBa51 cTopoHa H yrOJ1 npH OCHOBaHHH: AB 9, LC 15°. H;Jii
llHTe J1JJHHY BbICOTbl GH, npoBen.eHHoH K cTopoHe AB.

KOHTpOJlbHan pa60Ta no reMaM 4 H 5
63
5. B npHMoyroJlbHoM TpeyrOJIbHHKe ABC c npSlMblM yrJloM C H3BeCT-
ll.JIMHY K3TeTa AC.
BapHaHT 5
1. PaccMaTpHB351 non.o6Hble TpeyrOJIbHHKH
1BC H ABI C 1 , HaHJlHTe WHPHHY peKI1 BBI
(IJ MeTpax), eCJ1H AC 85 M, ACt 40 M,
/1B 1 50 M.
B
-----
---- ---------
-----
----
---
--B
--.- I
:. B paBHo6ellpeHHoH TpaneUHH ABCD H3BeCTHbi J]JIHHbI oCHoBaHHH
· \ D 39, BC 15, a TaK>Ke ,MHH3 60KOBOH CTOpOHbl AB 20. HaH-
u
/11 rre 11J1HHY BblCOTbI 3TOH TpaneUHH.
:c. (1 pasHo6eJlpeHHOM rpeyrOJIbHHKe ABC C OCHOBaHHeM BC 3allaHbi
'IJIIIJla 60KOBOH CTOpOHbl H }lJ1HHa BbICOTbl, npOBeJleHHOH K 60KOBOH
I I ()I)OHe: AB 29, C H20. HaHJ1HTe lL1IHHY CTOpOHbl BC, eCJIH 113-
IH" '-1'110, LITO yro.n npH BepwHHe A TynoH.
.1. 1\ TpeyrOJIbHHKe ABC CTopOHa AB paSHa 15, a cos LA 0,96.
11.li'I.JLfTe DJIHHY BbICOTbl, npoBelleHHoH K cTopoHe AC.
:. I  rlpHMoyro.nbHoM TpeyrOJIbHHKe ABC c npHMblM yrJIOM C "PO-
.". tll '11<) BblCOT3 C H. 113secTHo, QTO AC 8, AH 55. HaHJlMTe
· ""1. B.
BapHaHT 6
I It. It 'CM3TpHBaSl nOllOOHbIe TpeyroJlbHHKH
"'t' II ABIC l , H3i1D.HTeWHpHHypeKH BBI
· It ".'1 'II );lX), eCJIH AC 105 M, AC 1 50 M,
"I, .'15 M.
B
-_... -------
-----
-----
---
----
- - -:.B
- - - I

64
KOHTpOJlbHaH pa60Ta no reM8M 4 H 5
2. B paBH06eJlpeHHOH TpaneUHH ABCD H3BeCTHblllJIHHbi OCHOBaHHH
AD 98, BC 34, a TaK>Ke DJ]HHa 60KOBOH CTOpOHbl AB 68. HaA-
u
llHTe .MHHY BbICOTbl 3TOH TpaneUHH.
3. B paBHo6e.apeHHoM TpeyroJlbHHKe ABC C OCHOBaHHeM BC 3an.aHbl
JtJIHHa 60KOBOH CTOpOHbl H JIJIHH3 BbICOTbl, npoBelleHHoH K 60KOBOH
cTopoHe: AB 37, CH 12. HaHltHTe JtJIHHY CTOpOHbI BC, eCJlI1 H3-
BeCTHO, liTO yroJl npH BepWHHe A OCTpbIH.
4. B TpeyrOJ1bHHKe ABC JI.JIHH3 BblCOTbl BH, npoBeAeHHoH K CTOpO-
He AC, paSHa 24. HaHllHTe JI.JIHHY CTOpOHbl AB, eCJ1H H3BeCTHO, QTO
cos LA 0,1 91.
5. B npflMoyroJlbHoM TpeyrOJIbHHKe ABC c np5lMbiM yrJIOM C npOBe-
lleHa BbICOTa CH. H3BecTHo, QTO AC 16, CH 2 39. HaHD.HTe
sin LB.
BapHaHT 7
1. Ha cTopoHax AB 11 AC TpeyrOJIbHHKa ABC 83S1TbI COOTBeTCTBeH-
HO TOlJKH K H L T3K, lJTO npHMaSl K L napaJIJleJIbHa cTopoHe BC.
OTpe3KH CK H BL nepeceKalOTcSI B TOllKe P. HaHJlHTe OTHoweHHe
CP : PK, eCJlH H3BeCTHo, liTO AK: BK 3: 4.
2. Ha pHCYHKe H306pa>KeHa TpaneUHH, MH
KOTOpOH H3BeCTHbI !{JIHHbI ee CTOpOH AB,
BC, CD, a TalOKe JI)IHHa oTpe3K3 AH, rlle
H - OCHOB3HHe BbICOTbI BH 3TOH Tpane- A
llHH. Haii.QHTe JlJIHHy CTOpOHbI AD.
B
33
c
D
3. B paBHo6e.npeHHoM rpeyrOJIbHHKe ABC C OCHOBaHMeM BC H3BeCT"
Hbl MHHa 60KOBOH CTOpOHbl H .nJIHH8 OCHOB3HHH: AB 10, BC 13.
OTpe30K C H BbICOTa, npoBeJteHHaH K cTopoHe AB. HaH,lIJiTe MHHY
oTpe3Ka AH.
CTopOHe AC, paBHa 73. HaHJlHTe MHHY CTOPOHbl AB.

KOHTpOJIbHaR pa60Ta no reM8M 94 H 95
65
5. B npHMoyrOJIbHOM TpeyroJlbHHKe ABC c npHMblM yrnoM C npoBe-
neHa BbICOTa CH. 113BecTHO, liTO sin LA 0,4, AB 18. HaHJlHTe
JlJ1I1HY oTpe3Ka AH.
BapHaHT 8
I. Ha CTOpOH3X AB 11 AC TpeyrOJ1bHHKa ABC B3S1Tbl COOTBeTCTBeH-
110 TOqKH 1< 11 L T3K, liTO npHMa51 K L napaJIJIeJ1bHa CTopOHe BC.
()Tpe3KH CK H BL nepeceKalOTCSI B TOtlKe P. HaHHMTe OTHorneHHe
/1[(: BK, ec.nH H3BeCTHO, liTO CP: CK 13: 21.
2. I-Ia PHcYl1Ke H306pa)l(eH3 TpaneUH5I ABCD, lI/IH KOTOpOH H3BeCT-
IIIJI1{J1HHbl CTOpOH AB, CD H oTpe3KoB AH, DH, rn.e H OCHOBaHHe
I\lJICOTbl BH. HaHl!HTe D.JIHHY CTOpOHbl BC.
B
10
A H
43
D
:1. 1\ paBHo6ellpeHHOM TpeyrOJIbHHKe ABC C OCHOBaHHeM BC H3BeCT-
111.1 JlJIHHa 60KOBOH CTOpOHbl H JIJ]HHa OCHOB8HI1H: AB 25, BC 39.
I ) II )(\:JOK C H BbICOT3, npoBe.neHHafi K cTopoHe AB. HaHAHTe JIJIHHY
"III(':Ka AH.
I I 't I'f)eyro.nbHHKe ABC cTopoHa AB paBHa 52, a tg LA
1&11.' .l1.JlliHY BbICOTbI, npoBeJteHHoH K cTopoHe AC.
5 H u
12. aH-
'. I  II(1HMoyrOJIbHOM TpeyrOJIbHHKe ABC c npHMbIM yrJloM C npoBe-
I' ILl IU)ICOT8 CH. 113BecTHO, liTO cos LA 0,3, AC 9. HaHl{HTe
. 'fll'V ()Tpe3K3 BH.
BapHaHT 9*
, 'I" .f111arOHaJiH AC napaJIJ1eJ10rpaMMa ABCD B3S1T3 TOtJK3 P,
Ii Ict'll(c) nepeCeQeHHS1 npHMbIX BP HAD. HaHD.HTe OTHOllleHHe
t,. f ./), ec.nH H3BeCTHO, liTO AI< : DK 5 : 3.
I \t k _-I nO:lJ. K on TeM. TCCThl

66
KOHTpO.RbHaH pa60T8 no reMaM 4 H 5
2. Ha KJIeTlJaTOH 6YMare H306pa>KeH TpeyronbHHK B
ABC (CM. PHCYHOK cnpaBa). ClJHTaH, liTO CTopOHa
KneTKH paSHa 1, HaH!lHTe JlJIHHY MeHaHbI K CTOpO-
He BC. A
3. llJlHHbl OCHOBaHHH TpaneUHH paBHbl 11 H 36, a J1JIHHbI 60KOBbiX CTO-
pOH paBHbI 25 Ii 30. HawAiiTe J1J]HHY BbICOTbl 3TOH rpaneUHH.
4. B paBHo6ellpeHHoM TpeyroJlbHHKe ABC C OCHOBaHHeM AC
BaSI CTopOHa paBHa 21, a sin LC 0,6. OTpe30K AH BblCOTa
rpeyronbHHKa. HaHJlHTe JlJIHHY oTpe3K3 BH.
5. B npHMoyrOJ1bHOM TpeyroJlbHHKe ABC c npHMblM
Be,neHa BblCOT3 C H. 113BecTHo, 4TO AC 29, C H 21.
J1JIHHY KareTa BC.
BapH8HT 1 0*
1. Ha .QHarOHaJIIi AC napaJlJIeJlOrpaMMa ABC D B3S1Ta TOqK8 P,
Q - TOlJK3 nepeCeqeHHSI npSlMbIx BP H CD. HaHJlMTe OTHorneHHe
AP: CP, ec.nH H3BeCTHO, qro CQ: DQ 7: 2.
2. Ha KnerqaTOH 6YMare H306pa>KeH TpeyroJlbHMK
ABC (CM. PHCYHOK cnpaBa). CqHT3H, LITO CTopOHa
KJIeTKH paSHa 1, H3iiJ1HTe lIJIHHY MenHaHbI K CTOpO-
He BC.
3. J1HHbl OCHOBaHHH rpaneUHH paBHbl 7 H 35, a DJIHHbl 60KOBblX
pOH paBHbl 17 H 25. HaHIlHTe DJIHHY BbICOTbi 3TOH TpaneUHH.
4. B paBH06eJIpeHHOM TpeyroJlbHHKe ABC C OCHOB3HHeM AC -
BaSI CTopOHa paBH3 5, a sin LC 0,96. OTpe30K AH - BblCOT3
TpeyrOJ1bHHKa. HaH,QHTe DJIHHY oTpe3Ka BH.
5. B npSlMoyroJlbHoM rpeyrOJIbHHKe ABC c n pHMblM yrJIOM C
DJIHHY oTpe3Ka BHe

96. n.nOlllaJIb
67
 6. nJlOlQ.aAb
6.1. 3aAaI.JH Ha BblllHCJleHHSI
1. HaHHTe nJloru311b TpeyrO.flbHHKa, H306pa>KeH-
floro H3 KJIeTQaTOH 6YMare, HMelollleH pa3Mep KJ1e-
.rOK 1 CM X 1 CM. OTBeT 113HTe B KBan.paTllblX caHTH-
MeTpax.
 ....
 '--
 ,

2. HaHllHTe nJlOma,Ub TpeyrOJlbHHKa, H306pa>KeH-
11()rO Ha KJIeTtJaTOH 6YMare, HMelOllleH pa3Mep KJIe-
I ()K 1 CM X 1 CM. OTBeT n.3HTe B KBan.paTHbIX caHTH-
1c'rpax.
;1. HaHJ!HTe nJIOlllaJlb TpaneUHH, H306pa/KeHHOH
11;1 KJIeTlJaTOH 6YMare, HMelOllleH pa3Mep KJ1eTOK
IlM x lCM. OTBeT JlaHTe B KBaJlpaTHblX C3HTHMeT-
I. I X .
,1. 1-laHAHTe nJIOm311b TpaneUHH, Ji306pa)KeHHOH
11.1 I<JleTlJ3TOH 6YMare, HMelOllleH pa3Mep KJ1eTOK
II · 1 X 1 CM. OTBeT Jl3HTe B KB3,apaTHbIX C3HTHMeT-
I '.1 \ ·
 
 ,

:.. IlaHllHTe nnOlllal1b TpeyrOJIbHHK3, H306pa>KeH-
"I II f) II a KJleTlJ3TOH 6YMare, HMelOLl.{eH pa3Mep KJIe-
II -1\ I L M X 1 CM. OTBeT JlaHTe B KBallpaTHbIX caHTH-
""llt;IX.
,t 11;1 i1J1HTe nJlOlll3Jlb Tpeyro.nbHHKa, M306pa/KeH-
"1 -1'1 11;1 KJIeTQaTOH 6YMare, HMelOllleH pa3Mep KJIe-
, , '" I (M X 1 CM. OTBeT JlaHTe B KBallpaTHbIX caHTH-
...1 t 11.1\.

68
..
6. JIJ/OIll8JI.b
7. HaHJ1HTe nJIOlllaJ1b 4eTbIpexyrOJIbHHKa, H.306-
pa>KeHHOrO Ha KJIeTtlaTOH 6YMare, HMelOlUeH pa3-
Mep KJIerOK 1 CM X 1 CM. OTBer !laHl'e B KBallpaTHbIX
caHTHMeTpax.
8. HaHllHTe nJI01ll3)1b tleTbIpexyrOJ1bHHKa, H306-
paiKeHHoro Ha KJIeT4aTOH 6YMare, HMelOlUeH pa3-
Mep KJIeTOK 1 CM X 1 CM. OTBeT JlaHTe B KBaJlpaTHblX
caHTHMeTpax.
9. HaHJlHTe nJlourallb rpeyrOJIbHHKa, 11306-
.. "
pa*eHHoro B KOOp,UHH8THOH nJIOCKOCTH H3
n.aHHOM cnpaBa pHcYHKe.
4 -----------------
I
I
I
I
I
1 I
.-. -- - I
I I I
0 2 6 x
1 O. HaiiD.HTe nJlOlUab TpeyrOJlbHHKa, H306-
.. "
paJKeHHOrO B KoopHHaTHoH nnOCKOCTH H3
)t3HHOM cnpasa pHcYHKe.
---
2 --- I
I
I
I
2
o
8 x
11. HaMJlHTe nJIOlUa,llb 1JeTbIpexyrOJlbHHKa,
H306pa>KeHHOrO B KOOPJ1HH3THOH nJlOCKOCTH 4 ------
Ha n.aHHoM cnpaBa pHCYHKe.
1 ---
I I I
023
6 8 x
12. HaHD.HTe nJlOlUaJlb qeTblpexyrOJlbHHKa,
H306paJKeHHOrO B KOOpJ:lHHaTHOH nJlOCKOCTH
Ha JJ.aHHOM cnpaBa pHCYHKe.
-------------
3 ------
1 ------.
o 3 7_.

96. nJ/OUl8Jlb
69
13. HaHJlHTe nJT01ll311b TpeyrOJIbHHKa, H306-
.. "
paJKeHHOrO B KoopHHaTHoH nnOCKOCTH Ha
Ll3HHOM cnpaBa pHcYHKe.
3 ---
I
I ----:----1--- I
o 2 4 6
x
14. HaHllHTe nJlO1ll3Jlb rpeyrOJ1bHHKa, H306-
.. u
IJa}KeHHOrO B KoopnHH3THoH nJlOCKOCTH Ha
llaHHOM cnpaBa pMcYHKe.

5 --- I
I
I
I
I
2 ----:----
I I
1 -------1-------- I
o
2 4
8 x
15. HaHJlHTe nJIOlll3Hb qeTblpexyrOJlbHHKa,
IIJo6pa)KeHHOrO B KOOp,UMH3THOH nJ10CKOCTH
11;lllaHHOM cnpaB3 pHcYHKe.
3
2 ---
I
o
I
I
I
-- -------
I
----I--t-_-.
I I
I
2
4
6
8 x
I (). HaA,UHTe nJIOlUaJ1b lJeTblpexyronbHHKa,
II u)Opa*eHHoro B KOOpn.HH3THOH nJIOCKOCTH
11.1 Jl,IHHOM cnpaBa pHcYHKe.
----------------
1 -
012
I
I
I
I
I
I
I
I
I
I
3
7 8 x
I ti. l;aHllHTe nJIOllla,llb napaJlJleJ10rpaMMa, eCJJH llJIHH3 O-UHOH 113 ero
· ,. 'I )()I-I paSHa 9, llJIHH3 pyroH paBHa 12, a OllHH 113 yrJJOB paBeH 30°.
I H. I IaHllHTe nJlOLUa)lb napanJIeJ10rpaMMa, eCJlH llJIHHa OD.HOH 1-13 era
IIII )()II paSHa 5V3, llJIHH3 llpyroH paSHa 8, a O)1HH H3 yrJIOB paBeH 60 0 .
It'. I I (I HllHTe nJIOlllal1b pOM6a, eCJIH ero CTOpOHbl paBHbl 15, a OJlHH H3
. I II' .1' l>aBeH 150 0 .
11()IillHTe nJIOlU3Jlb TpeyrOJlbHMKa, CTOPOHbl KOToporo paBHbl 6V2
 I yroJl Me)KLlY 3THMH cTopoHaMH p3BeH 135 0 .

70. 6. nJ10I1lab
21. HaHJ1HTe nJlOlU8J1b TpeyrOJlbHHKa, CTOpOHbl KOToporo paBHbI 4V3
H 12, a yroJl Me}l{.ny 3THMH CTOpOH3MH paBeH 120 0 .
22. I-IaHJlHTe ltJIHHY CTOpOHbl pOM6a, eCJ1H era nJIOlllallb paSHa 32,
a OD.HH H3 yrJ10B paBeH 45°.
23. HaCIJlHTe rpallYCHYIO Mepy oCTporo yrJIa pOM6a, eCJIH ero nJ10mallb
pasHa 72, a JlJ1HHa CTOpOHbI pasHa 12.
24. HaHJlHTe rpallycHYIO Mepy Tynoro yrJl3 napa.nJIeJIOrpaMMa,
ero nJlOm3)lb pasHa 60, a llJIHHb1ltBYX era CTOpOH paBHbI 8 H 15.
25. 4eMY MO}KeT 6bITb paBHa rpan.ycHa51 Mepa yrJIa npH
paBHo6en.peHl10ro TpeyrOJlbHHK3, eCJ1H H3BeCTHO, tITO ero nJ10man.b
Ha 25, a llJIHH3 60KOBOH CTOpOHbI paBHa 10?
26. 4eMY MO>KeT 6blTb paBHa rpaYCHa51 Mepa yrJl3
pasHo6e,QpeHHoro TpeyrOJ1bHMKa, eCJIH H3BeCTHO, l
11a 98, a llJIHHa 60KOBOH CTOpOHbI pasHa 14?
27. HaHllHTe nJlOmall.b paBH06eJlpeHHOro TpeyrOJIbHHK3, eCJlH
ero 60KOBOM CTOpOHbI paBHa 25, a !lJIHH3 OCHOBaHHH pasHa 48.
28. HaHllHTe nJIOmallb paBH06e!lpeHHOro Tpeyro.nbHHK3,
ero 60KOBOH CTOpOHbl paSHa 26, a !lJIHHa BbICOTbl, npOBelleHHOH
110BaHHIO, pasHa 24.
29. HaHllHTe llJlHHY 60KOBOH CTOpOHbl paBHo6ellpeHHoro ·
Ka, eCJlH era nJlOlllallb paBH3 480, a AJIMH3 OCHOBaHHH paBHa 32.
30. HaHJlHTe DJIHHY 60KOBOH CTOpOHbI I
Ka, eCJIH era n.no1ll3,Llb paSHa 420, a AJIHH8 BbICOTbI, nposelleHHOH K
HOB3HHIO, paBHa 35.
31. HaHD-HTe nJlOma,nb pOM6a, eCJIH ero ,QHarOHaJIH paBHbI 12 H
Ii
32. HaH!lHTe nJlOlUaAb pOM6a, eCJlH ero ,[lHarOHaJlH paBHbl 11 H
33. nJlO1ll3J1b pOM6a pasHa 135, a O,UHa H3 era J1HarOHaJIeH
HaHLlHTe ltPYrylO JlHarOHaJIb pOM6a.

6. nJlO1ll8J1.b
71
34. nJlOLUa.nb pOM6a paBHa 110, a OD.Ha "3 ero JlHarOHaneH paBHa 10.
HaH,QuTe llPYrylO ,UHarOHaJlb pOM6a.
35. nJIOLUaD.b pOMoa paBHa 840, a OllHa H3 ero llHarOHaJIeH paBHa 42.
HaHD.HTe nepHMeTp pOMoa.
36. nepHMeTp pOM6a paBeH 164, a 0J1H3 H3 ero JlHarOH3JIeA paBH3 80.
Hal1HTe nJlOLUa)lb pOM6a.
37. HaHDJiTe nepliMerp npS1MoyronbHHKa, eCJIH era nJlOLUa,Ub paBHa
25, a OD.Ha H3 CTOpOH pOBHO B lJeTblpe pa3a 60JIbWe .npyroH CTOpOHbl.
38. HaHJlHTe nepHMerp npSlMoyrOJlbHHKa, eCJ1H era nJlOLUa.llb paBH3
192, a OTHOllleHHe JlJIHH coce,nHHX CTOpOH paBHo 3 : 4.
39. HaHllHTe nepHMeTp npHMoyrOJlbHHK3, eCJ1H ero nnOLUa,llb paBHa
126, a pa3HocTb Me)KJIY 60JlbWeH H MeHbweH cTopoHaMH paBHB 5.
40. HatiJlHTe nepHMeTp npSlMoyrOJIbHHKa, eCJIM ero nJIOLUa.Qb paBHa
:28, a pa3HocTb Me)KJlY 60JlbWeM 11 MeHbweH cTopoHaMH paBHa 7.
4 1. HaHJ1HTe nJIOLU3,llb TpaneUHH, eCJIli JlJIHHbl ee oCHoBaHHH paBHbl 17
II 25, a BblCOTa paBHa 9.
.12. 011HO 113 OCHOBaHHM TpaneLlHH pasHo 27, BblCOTa paBH3 8, a nJIO-
1I1:lllb pasHa 176. HaHllMTe BTopoe OCHOBaHHe 3TOH TpaneUHH.
,I:J. BbICOT3 TpaneLlHH pasHa 6, a nJlO1ll311b paBHa 192. HaHJlHTe MMHY
u u
· I Jt 'Jl11eH JIM"H" 3TOM Tpaneu."H.
.1,1. CpeHSlH JIHHHSI TpaneI.J.HH paBHa 22, a nJlOI..Ll311b paBHa 352. H3H-
u
till It \ AbIcory 3TOH rpaneUHH.
.1 r). OCHosaHHH paBHo6e.npeHHoH TpaneUHH paBHbl 8 H 50, a ee nJlo-
111.1 '11) pasHa 580. HaHllHTe DJIHHY OOKOBOH CTOpOHbl 3TOH Tpaneu.HH.
1ft. ()cHoBaHHSI paBH06eJ1peHHOU TpaneLlHH paBHbl 22 H 58, a ee "JlO-
u." 'II. IJaSH3 960. HaHD.HTe nepHMeTp 3TOH TpaneUHH.
'1 ,. ()CHOB3HHSI paBH06eJlpeHHOH TpaneUH" paBHbl 23 Ii 47, a ee ne-
'."'-1.' II) paBeH 96. HaHllHTe nJIOlU8n-b 3TOH TpaneUHH.

72
"
 6. n JI 0 maJ].b
48. OCHoBaHHH paBH06eJlpeHHOH TpaneUHH paBHbl 18 H 34, a ee ne-
pHMeTp paBeH 86. HaHJtHTe nJI01U311b 3TOH TpaneUHH.
49. nepHMeTpbl ,llBYX non.o6HbIX MHoroyrOJIbHHKOB OTHOCs:tTC5I KaK 3 : 5,
a nJIOlUaJI.b 60JIbWerO H3 HHX paBHa 30. Hai1D.HTe nJIOlllaJlb MeHbwero
MHoroyroJlbHHKa.
50. nepHMeTpbI JI.BYX noo6HblX MHoroyrOJIbHHKOB OTHOCSlTC5I KaK 4 : 7,
a nJlOllla)!b MeHbwero 113 HHX paBHa 40. HaiiD.HTe nJIOllla-ab 60JlbWerO
MHoroyroJlbHHKa.
51. nJlOLUa.QH ABYX nOJl06HbIX MHoroyrOJ1bHHKOB OTHOCSlTCSI KaK 9 : 16.
nepHMerp 60JIbWerO MHoroyro.nbHHKa paBeH 28. HaHllHTe nepHMeTp
MeHblllero MHoroyroJlbHHKa.
52. nJ10111aJlH .nBYX nOllo6HbIX MHoroyrOJIbHHKOB OTHOCHTCfI KaK 4 : 25.
nepHMeTp MeHbwero MHoroyrOJIbHHK3 paBeH 11. HaHJlHTe nepHMeTp
60JIbWerO MHoroyrOJIbHI1Ka.
53. nJlOLLlaJlb TpeyrOJIbHHK8 ABC pasHa 76, oTpe30K MN CpeJlHSlSI
J1HHHH TpeyrOJIbHHKa, napaJlJIeJIbHa51 cTopoHe AB. HaHllHTe nJlO1ll8}lb
TpeyrOJIbHHKa CMN.
54. OTpe30K MN CpeJlliHH JlHHHH TpeyronbHHKB ABC, napaJ1J1eJ1b-
H3S1 cTopoHe AB. nJIOmallb TpeyrOJIbHHKa CMN pasHa 16. HaHJlHTe
nJIOLLla)!b TpeyrOJIbHHKa ABC.
55. nJIOLLlaJlb TpeyroJlbHHK3 ABC pasHa 45. Ha cTopoHax AC H BC
B3S1TbI T04KH M H N TaK, liTO orpe30K MN napa.nJleJIeH CTopOHe AB,
npHlJeM MN : AB 2 : 3. HaMJlHTe nJlOlllanb TpeyrOJIbHHKa CMN.
56. Ha CTOpOH8X AC, BC TpeyrOJIbHHKa ABC B3HTbi COOTBeTCTBeH-
HO TOllKH !vI, N TaK, liTO AAI : CM BN : CN 4 : 5. OJlOLLlaLlb rpe-
yroJlbHHKa CMN paBHa 15. HaHJlHTe nJI01ll3D.b TpeyroJlbHHKa ABC.
57. nJlOLLl3Jlb TpeyrOJIbHHK3 ABC paSHa 75. Ha cTopoHe AB 83HT(1
T04Ka M T3K, liTO AM : BM 3 : 7. HaHllHTe nJ101113Ab TpeyrOJ1bHHKcl
AMC.

6. nJlOl11ap.b
73
58. Ha CTopOHe AB TpeyroJlbHHK3 ABC B3HTa TOLJK3 M T3K, liTO
AM: BM 5 : 8. nJlOmallb TpeyroJlbHHKa AMC pasHa 10. HaHJlHTe
nJIOm3J{b TpeyrOJIbHHKa ABC.
59. n.nolUa.nb TpeyrOJIbHMK3 ABC pasHa 100. Ha cTopoHax AC M
BC B3HTbi COOTBeTCTBeHHO TOqKH M Ii N TaK, liTO AM: CM 8 : 7,
BN: CN 2 : 3. HaH!lHTe nJlOlUaJtb TpeyrOJIbHHKa CMN.
60. Ha cTopoHax AC H BC TpeyroJlbHHKa ABC B3S1TbI COOTBeT-
CTBeHHO TOllKH M 11 N T3K, lITO AM: Ck! 3 : 4, BN: CN 5 : 6.
l-Iai1D.HTe nJ10ll.la,nb TpeyrOJIbHHKa ABC, eCJIH nJlOlUa.Qb TpeyroJlbHHK3
CMN paSHa 72.
B 3aLlalJ3X 61 - 66 CJIeJlYlOlU3S1 lJ3CTb YCJlOBHH SlBJIHeTCH 06ll.leH:
(BHyrpM npHMoyrOJIbHHK3 ABC D B3SIT3 TOLJK3 P. 4epe3 3TY T04KY
IlpOBeJleHbI npHMbIe, napaJIJIeJ1bHbIe cTopoHaM npHMoyroJlbHHKa M ne-
I )CCeKalOlUHe HX B TOIJKaX K, L, M, N. »
B
L
C
M
() 1. 113BecTHbI nJIOma)tH npSlMoyroJlbHHKOB
.11(PN, BKPL, CLPM CM. J!aHHbIH cnpa-
1';1 PHCYHOK. HaHJlHTe nJIOlUa,Qb npHMoyroJlb-
IlIlKa DNPM.
K 24 P 48
36 ?
.
fi2. 113BecTHbi nJIOlUa,UH npHMoyrOJIbHHKOB
\ I\"PN, DMPN, CLPM - CM. n.aHHbIH cnpa-
1'.1 I )liCYHok. HaHJI.HTe nJIOma,llb npSlMoyrOJIb-
11111,;1 ABCD.
A
B
N
L
D
C
M
K -P 30
16 40
A
N
L
D
fi:C. 11JlOlllallb npSlMoyrOJIbHHKa ABCD paB-
ILl :()O. H3BeCTHbl nJlOlUaJlH npSlMoyrOJIbHH- K
I'. II' nNPM H CLPM CM. J{3HHblH cnpaB3
1'11 1 'YIIOK. HaH.nHTe nJIOma)1b npSlMoyroJlbHH-
I !I Il/(PL.
B
? 80
. P
120
c
M
A
N
D
'II. IIJIOmaD-b npHMoyrOJIbHHKa ABCD paSHa 90. H3BecTHbI nJlOlUa-

74
.aM npSlMoyronbHMKOB AKPN H BKPL, CM.
llaHHblH cnpaBa pI1CYHOK. HaMllHTe nJ10111anb
npHMoyroJlbHHKa DNP/Jtf.
B
K 12 P
15 ?
.
 6. nJlOI1laJl.b
L C
M
A
N
D
C
65. nJlOwa,llb npHMoyroJlbHHK3 ABCD
paBH3 207. H3BecTHbI nJIOwaJlH npHMo-
yroJlbHHKOB AKPN H CLPM CM. 113H-
HblH cnpaB3 PHCYHOK. 4eMY MO>KeT 6blTb
paBHa nJlOwallb npHMoyrOJIbHHKa BKPL?
B L
? p 7S
32
A N
M
D
L C
20
M
66. nJlO1ll3Hb npHMoyrOJIbHHK3 ABCD
paBH3 135. H3BecTHbI nJlOwaD-H npHMo-
yroJlbHHKOB AKPN H CLP!vf CM. JlaH-
HbIM cnpaBa PHCYHOK. 4eMY MO>KeT 6blTb
pasHa nJIOma)lb npS1MoyrOJIbHHKa DMPN?
B
50
?
.
A
N D
67. -aBe npOTHBOnOJlO)KHble CTOpOHbl KBaJ1pa-
u u
T8 pa3JleJIeHbi Ha weCTb paBHblX qaCreH, KpaHHHe
T01.JKH JteJIeHHSI COeJlHHeHbI HaKpeCT OTpe3K3MH Ii
qaCTb KBaJtpaTa, orpaHM t leHH8H 3THMH OTpe3KaMH,
u
33WTpHxoBaH8 CM. JlaHHbIH cnpaBa PHCYHOK.
a) nJIOLUa.Qb Bcero KBaJlpaTa paBHa 69. HaHllHTe nJlOllla,llb 3awTpHxo-
u
BaHHOM lJaCTH KBaJ{paT3.
6) nJIOwa.nb 3awrpHxosaHHoH ifaCT" KBa,apara paBHa 243. HaHJlHTe
DJIHHY CTOpOHbI KBallpaTa.
.
I
68. ,UBe npOTMBOnOJIO)KHble CTOpOHbl na-
paJIJIeJIOrpaMMa pa311eJIeHbl H3 TP" paBHble
u
qaCTH, KpaHHHe TOqKH eneH"H coeHHe-
Hbl H3KpeCT OTpe3K3MH M qaCTb napaJlJIe-
norpaMMa, orpaHHtJeHHaH 3THMH OTpe3Ka-
u
MH, 3aWTpHXOBaHa CM. ,118HHbIH cnpaBa PHCYHOK.
a) nJIOLUaJlb Bcero nap3JIJIenOrpaMMa paBH8 228. HaHAHTe nJI01ll3JUI

6. nJ/Omap.b
75
u
33WTpHXOBaHHOH (laCT" napaJIJleJIOrpaMM3.
6) nJlOma,Qb 3awTpHxoB3HHOH LJ3CTH napaJlJIeJIOrpaMM3 pasHa 14.
l-IaHJlHTe BbICOTY napaJ1J1eJIOrpaMMa, ec.nH ero CTopOHa pasHa 12.
69. 8 napaJIJIeJlOrpaMMe ABCD H3 llHarOHaJlH
AC B3HTbl TOllKH K H L TaK, liTO AK O,3AC,
C L O,2AC, CM. PHCYHOK. HaHJ1HTe nJIOUlallb
lIeTblpexyrO.flbHHKa BLDK, eCJIH nJ101113Hb na-
paJIJ1eJIOrpaMM3 ABCD paBHa 69.
A
c
70. B napaJIJIeJ10rpaMMe ABCD Ha ll H a rO H3JIH
(} L 1;)2 AC, CM. PHCYHOK. HaH!lHTe nJlOlllaJlb
'ICTblpexyrOJIbHHK3 BLDK, eCJ1H nJIOlllan.b na-
I );IJ1JIeJ10rpaMMa ABCD pasHa 168.
A
c
71.* B napaJlJleJIOrpaMMe ABCD Ha llHarOHa-
,IJSIX AC H BD B3HTbi COOTBeTCT B eHHO n a pb I TO-
I ; ,\..[ DNA
11.11<)lllab tleTblpexyroJlbHHK3 KMLN, eCJIH nJIOlllaJlb napaJ1J1eJ10rpaM-
11 ABCD paBHa 108.
c
/2.* B napaJIJIeJIOrpaMMe ABCD Ha llHarOHa-
'lUX AC H BD B3SITbI COOTBeTCT B e HH O napbl T O-
'. 1 1
.,. AC, BlYf 6 BD, DN 3 BD, CM. PH- A
I \' Ile)K. HaHJ{HTe nJIOUl3.Qb lJeTblpexyroJlbHHKa I(MLN, eCJIH nJIOllla,l1b
'LII)IJI.1JeJ10rpaMMa ABCD paBHa 14.
c
1:1. 'IOIIKH 1(, L, M, N - cepeJJJiHbl CTOpOH npSlMoyronbHHKa ABCD,
" 'lh1 P npHHallJle>KHT OTpe3KY KL, CM. PHCYHOK K CJIenylOlileH 33,lta-
" II ;11fJlHTe nJIOlllallb TpeyroJlbHI1Ka MNP, eCJIH nnOlllaJI.b npHMO-
· t .. .tll.III<IK3 ABCD p3BHa 15.

76
..
96. nJ10maJ1.b
74. TOlJKH K, L, M, N - CepeJlHHbl CTOpOH npH- B
MoyrOJIbHHKa ABCD, TOtlKa P npHH3D.JJe>KHT OT-
pe3KY KL, CM. PHCYHOK. HaHJtHTe nJ10mallb npH-
MoyrOJIbHHKa ABCD, eCJlH nJIOmallb TpeyrOJ1b- A
HHKa MNP pasHa 47.
75.* I-Ia cTopoHax qeTblpexyrOJIbHHKa ABCD
AK AN
B3HTbi TOllKH K, L, AI, N raK, liTO BK DN
CL CM 1 u
BL DAf 2 ' CM. PHCYHOK. HaMl!HTe nJlo-
llla)lb 4eTblpexyr0J1bHHKa KLAfN, eCJIH nJIOllla}lb
lleTblpexyrOJIbHHKa ABCD pasHa 117.
76.* Ha CTopOHax 4eTblpexyr0J1bH HK3 A B CD
CL CM 5 u
BL DM 7 ' CM. PHCYHOK. HaHJ!HTe nJ10-
lllallb qeTblpexyro.nbHHKa ABCD, eCJ1H nJIOlllaJlb
lJeTbIpexyroJlbHHKa KLAtfN pasHa 77.
77 .* Ha KJIeTLJ3TOH 6YMare c pa3MepoM KneTOK
1 CM x 1 CM 306pa)l{eH nJiaH rOpOJlCKOrO cKBepa,
CM. PHCYHOK (TeppHTopHSI CKBepa 06Be.aeHa )l{Hp-
HOM JlHHHeH). OnpelleJIHTe MaCWTa6 3Toro H306pa-
)l{eHHS1, eCJIH H3BeCTHO, 4TO nJIOma.Llb CKBepa co-
CTaBJIHer 60 ra (1 fa 10000 M 2 ).
78.* Ha KJIeTlJaTOH 6YMare c pa3MepOM KJleTOK
5 MM X 5 MM 11306pa>KeH nJIaH npOJlOBOJIbCTBeH-
HorD pblHKa, CM. PHCYHOK (TeppHTopHH pblHKa 06-
Be!leHa >KHpHOH JII1Hlieti). OnpelleJIHTe MacwTa6
3Toro H306pa>KeHHH, eCJIH H3BeCTHO, liTO nJIO-
lUallb PbIHK3 paBHa 1,8 ra (1 ra 10000 M 2 ).
L
C
M
N
1)
N
A
N

6. nJlOmaJI.b
77
79.* B TpeyroJlbHHKe ABC Ha cTopoHax AB, AC H BC B3ffTbi COOTBeT-
CTBeHHO TOlJKH M, N H K TaK, liTO lIeTblpexyrOJIbHHK AMKN HBJlSI-
TpeyroJlbHHKa ABC. HaHllHTe !lJ1HHY CTOpOHbI AC, eCJIH H3BeCTHO, liTO
MK 6.
80.* B TpeyrOJIbHHKe ABC H8 cTopoHe AB B3HTbI T04KM M H N, a
Ha cTopoHax BC Ii AC B3S1Tbl T01.JKH P H Q TaK, liTO lJeTblpexyroJlbHHK
lvf NPQ HBJlHeTCH napaJl.fleJIOrpaMMOM, nJlOma,ab KOToporo COCT3BJlHeT
H3BeCTHO, liTO M N 1.
6.2. 3aAalfM Ha OKa3aTeJlbCTBO
81. Ha cpeJlHeii JlHHMH TpaneUHH ABCD C OCHOBaHI15IMH AD H BC
BbI6paJlH npOH3BOJIbHO TOllKY !vI. llOK3IKHTe, liTO CYMMa nJIOLU3JleH rpe-
yrOJIbHHKOB ABJv! H CD !vI paSHa nOJIOBHHe nJlOlllal{H TpaneUMH.
82. BHyrpH napaJlJIeJlOrpaMMa ABCD Bbl6paJlH npOH3BOJlbHO TQ1IKY
1\1. ,UoKa)KHTe, 4TO CYMMa nJlOm311eH TpeyroJlbHHKOB ABM H CD Al
pasHa CYMMe nJlOlllalleH TpeyrOJ1bHHKOB BC M " AD l\;[.
83. HHarOHaJlH tieTbIpexyrOJIbHHK3 ABCD nepeceKalOTCSI B TOtIKe P.
HOKa>KHTe, liTO npOH3BeJleHHe nJlOlU311eH TpeyrOJIbHHKOB ABP H CDP
paBHO npOH3BelleHHIO nJIOlUaJleH rpeyronbHHKOB BCP M ADP.
84. ,llHarOH3JIH TpaneUHH ABCD C OCHOB3HHSlMH BC H AD nepece-
KalOTCSI 8 TOlJKe P. lloK3)1{HTe, liTO nJIOmaD-H TpeyrOJIbHHKOB ABP H
CDP paBHbI.
85. ,UHarOHaJIH lJeTbIpexyrOJIbHHK3 ABCD nepeceKalOTCH B T04Ke P.
HOKa)KHTe, qTO eCJIH nJlOlUaD-H TpeyroJlbHHKOB ABP 11 CDP paBHbI,
TO CTOpOHbl BC H AD napaJlJleJIbHbl.
86. T04Ka M - cepe,QHH3 CTOpOHbl AB 1.JeTblpexyrOJIbHHK3 ABCD.
1!oKa)l{HTe, liTO nJIOLUallb TpeyroJlbHHK3 M C D paBHa nOJlYCYMMe nJIO-
m311eH TpeyrOJIbHHKOB AC D H BCD.

78
 6. n.l101118JI.b
87. TOlJK3!vI epeD.HHa CTOpOHbl AB LJeTblpexyrOJIbHHKa ABCD.
nJI01ll3Ab TpeyrOJIbHHKa lvICD paBH3 n0J10BHHe nJIOma,nH 4eTblpex-
yroJlbHHKa ABCD. l1oK3)1{HTe, liTO npHMble BC H AD napaJlJIeJ1bHbl.
88. nYCTb ABCD - TpaneUHH C OCHOBaHHHMH BC HAD, S - nJIO-
m3)tb 3TOH TpaneUHI1, d - paccToHHHe OT cepe,nHHbI 60KOBOH CTOpOHbl
AB llO npHMoH CD. 110 Ka)l{HTe, liTO S d · CD.
89.* Ha CTopOHe AB Bb InYK.n Or O lJeTbIpexyrOJIbHHK3 ABCD B3RTa
0603Ha4HM 4epe3 SA, Sa H SK 11JlOlllallH TpeyrOJIbHHKOB ACD, BCD
1 k
I-I !(CD COOTBeTCTBeHIIo.lloK3>KHTe, liTO SJ( SA · 1 + k + BB · 1 + k ·
90.* Ha 60KOBbiX CTOpOH3X AB H CD rpaneUHH ABC D B3HTbI co-
OTBeTCTSel-illO TO(IKH P H Q TaK, liTO nJIOlllanJi TpeyrOJIbHHKOB CDP
H ABQ paBHbI. lloKa/KHTe, liTO orpe30K PQ napaJ1J1eJ1eH OCHOBaHHHM
rpaneUJ1"1.
KOHTpOJlbHaSl pa60Ta.
PeKOMeHJlyeMoe BpeMH ,LlJlH BblnOJJHeHHSI pa60TbI 30 MHHYT.
Pel{OMeHJlYMaH WK3J13 OLleHHsaHHSI pe3YJIbTaToB:
'-' 0-1 2-3 4 5
ylt1cno BepHblX peweHVI
WKonbHa ou.eHKa 2 3 4 5
BapHaHT 1
1. HaHHHTe nJI01ll3Jlb TpeyrOJIbHHKa, H306pameH-
Horo Ha KJIeTlJaTOH 6YMare, HMelOllleH pa3Mep K.ne-
TOK 1 CM X 1 CM. GrBer llaHTe B KB3JlpaTHbIX C8HTH-
Merpax.
2. HaHlI.HTe nJIouraJ1b napanJIeJlOrpaMMa, eCJII1 D.JlHHa O,QHOH H3 ero CTO-
POlo' paSHa 13, D.J1HI-13 D.pyroH paSHa 17, a OIlMH 113 yrJ10B p3BeH 150 0 .

KOHTpOJIbHaa pa60ra
79
3. HaHJlHTe nepHMeTp npS1MoyrOJlbHHKa, eCJlH ero nJ10lU3)lb paSHa 45,
a OJl.Ha 113 CTOpOH pOBHO B nSlTb pa3 60JIbllJe .npyroH CTOpOHbl.
4. O-UHO "3 OCHOBaHHH TpaneUHM paBHo 14, BblCOT3 paBHa 6, a nJlo-
lUa,llb pasHa 90. HaHD.HTe BTopoe OCHOB3HHe 3TOH TpaneUHH.
5. nepHMerpbl JlBYX no,no6HblX MHoroyroJlbHHKOB OTHOCSITCH KaK 4 : 5,
a nJIOlUa,nb 60JlbWerO H3 HHX paBHa 20. HaHJtHTe nJlOlU3)lb MeHblllero
MHoroyrOJIbHHKa.
BapHaHT 2
1. HaHJlHTe nJIOUlallb TpeyrOJlbHHKa, H306pa>KeH-
Horo Ha KJIeTqaTOH 6YMare, HMeJOllleH pa3Mep KJIe-
TOK 1 CM X 1 CM. OTBeT .naHTe B KB8!lpaTHbIX caHTH-
Merpax.
2. HaH,llHTe nJlOUlall.b napaJlneJIOrpaMM3, eCJ1H J1JIHHa OJtHOH H3 era CTO-
pOH pasHa 12, JlJIHHa llpyroH paBHa 21 y'2, a O,QHH H3 yrJloB paBeH 135 0 .
3. HaH!lHTe nepHMeTp npSlMoyrOJlbHHKB, eCJIH era nnOll1an.b paBHa 90,
a O,llHa H3 CTOpOH B 3,6 pa33 MeHblUe J!pyroH CTOpOHbl.
4. OAHO H3 OCHOB3HHH TpaneLl"H paBHo 11, BblCOT3 paBHa 9, a nJIO-
ll1a,Ub pasHa 144. HaHllHTe BTopoe OCHOB3HHe 3TOH Tpaneu.HM.
5. nepHMeTpbl,nBYX no,nooHblX MHoroyronbHHKoB OTHOCSITCH K3K 4 : 7,
a nJIOUla,l1b MeHblllero H3 HHX paBHa 48. HaH,QHTe nnOll1a.nb 60JlbllJerO
MHoroyroJlbHHKa.
BapHaHT 3
1. HatillHTe nJIOmaLlb qeTblpexyrOJIbHHKa, H306-
pa>KeHHOrO Ha KJIeTQaTOH 6YMare, HMelOll1eH pa3-
Mep KJleTOK lCM x lCM. OTBeT J1aHTe B KB3.npaTHblx
caHTHMeTpax.

80  6. nJ/omaJI.b
2. HaH,ltHTe nJlOmaJlb TpeyrOJlbHI1Ka, CTOpOHbl KOToporo paBHbl 14-/3
H 7, a yroJl Me}KllY 3THMH cTopoHaMH paBeH 60°.
3. HaHJlHTe nepHMeTp npHMoyrOJlbHIiIK3, eCJIH ero nJlOma,ltb paSHa 250,
a OTHoweHHe ,lJJIHH cocellHHX CTOpOH paBHo 5 : 8.
4. BblcoTa TpaneUHH paBH3 6, a nJIOm3!lb paBHa 17 4. HaHJlHTe llJIHHY
u u
cpe,nHeH .flltH"H 3TOH TpaneLlHH.
5. nJlOlU31111 llBYX n0J106HbIX MHoroyrOJ1bHHKOB OTHOC5JTC5I KaK 4 : 49.
nepHMeTp OOJlbWerO MHoroyrOJ1bHHKa paBeH 105. HaHjlHTe nepHMeTp
MeHbwero MHoroyrOJIbHHKa.
BapHaHT 4
I. HaHJlHTe nJ10m3l!b 4eTbIpexyrOJlbl-lHKa, H306-
pameHHoro Ha KJ1eTlJaTOH 6YMare, HMelolUeCt pa3-
Mep KJ1eTOK 1 CM x 1 CM. OTBeT Jl3HTe B KBallpaTHblX
caHTHMeTpax.
2. HaHJlHTe nJlO1ll311b TpeyrOJlbHHKa, CTOpOHbl KOToporo paBHbI 3J2
H 15, a yroJl Me>KD-Y 3THMH cTopoHaMH paBeH 45°.
3. HaHJlHTe nepHMerp npHMoyrOJlbHHK3, eCJIH ero nJlO1l.la,llb paSHa 63,
a OTHoweHHe ATIHH coceJIJ-IHX CTOpOH pasHo 4 : 7.
4. Cpe.aHHSI JlHHHSI TpaneUHH pasHa 34, a nJI01l.la,Ub pasHa 221. HaHHH-
v
Te BbIC01)' 3TOH TpaneUHH.
5. nnOlllaD-H l{BYX no,ao6HblX MHoroyrOJIbHHKOB OTHOCHTCH K3K 9 : 25.
nepHMeTp MeHblllero MHoroyrOJIbHHKa paBeH 12. HaH.QHTe nepHMeTp
60JlbWerO MHoroyrOJIbHHKa.

KOHTpOJ1bHan pa60ra
81
BapHaHT 5
I. HaHllHTe nJIOlUaJlb TpeyrOJIbHHKa, 11306-
.. u
pa*eHHoro B KoopHHaTHoH nnOCKOCTH Ha
J{aHHOM cnpaBa pHcYHKe.
4 -------- I
3 ---
I
I
I
I I
1 ----:------- I
o 2 4 6 x
2. HatillHTe llJIHHY CTOpOHbl pOM6a, eCJlH ero nJIOlllallb paBH3 72V3, a
OJtHH 113 yrnoB paBel-l 120 0 .
3. HaHllHTe nepHMerp npHMoyrOJIbHHKa, ecnH ero nJIOlllaJlb paBH3 40,
a pa31-10CTb )tJlHI-I 60JIbWeH H MeHbweH CTOpOH pasHa 6.
4. OCHoBaHHH paBHo6eJtpeHHoH Tpanel.!HH paBHbI 14 H 30, a ee nJIO-
lU3)tb paBH3 330. HaHJ!I1Te nepHMeTp 3TOH TpaneUHH.
5. nJlOma,nb TpeyrOJIbHHKa ABC paBHa 50. Ha cTopoHax AC H BC
B3S1TbI TOlJKH M H N raK, liTO OTpe30K MN napaJIJleJleH CTopOHe AB,
npHlJeM MN : AB 3 : 5. HatiJ!HTe n.nOUlaHb TpeyrOJIbHHKa CMN.
BapHaHT 6
1. HaHJlHTe nJIOUlaJlb TpeyrOJlbHHK3, H306-
.. "
pa)KeHHOrO B KoopHHaTHoH nnOCKOCTH Ha
.uaHHOM cnpaBa pHcYHKe.
---------
2 ---
I
1 ---i-------r-- I
o
2
5 7 x
2. HaHJlHTe CHHYC yrJIa pOM6a, eCJlH )].JIHHa ero CTOpOHbl paSHa 18, a
nJIOma,llb paBH3 81.
3. HaHLlHTe nepHMeTp npSIMoyrOJIbHHKa, eCJII1 ero nJIOma)1b pasHa 176,
a pa3HOCTb J1J1HH 60.nbUJeH Ii MeHblUeH CTOpOH paBHa 5.
4. OCHoBaHHH paBH06eJlpeHHOH TpaneUHH paBHbl 18 H 32, a ee nJIO-
Ul3Jtb p3BHa 600. H3ti.nHTe nepHMeTp 3TOH TpaneUHH.

82  6. nJlOIll8Jlb
5. Ha CTopOHax AC " BC TpeyroJlbliHKa ABC B3HTbI TOlJKH M 11 N
TaK, liTO OTpe30K MN napaJIJleJleH CTOpOHe AB " MN: AB 5 : 6.
nJIOllla!lb TpeyrOJIbHHKa CMN paBHa 20. HaHJlHTe nJlOlllaJlb TpeyroJlb-
HliKa ABC.
BapHaHT 7
1. HaHJlHTe nJloru8llb qeTblpexyrOJIbHHKa,
H306pa)KeHHOrO B KOOpJ1HHaTHOH nJlOCKOCTH
H3 -n3HHOM cnpaB3 pHcYHKe.
5 -------- I
3 --- I
I
2 ----.---- -------
I I I
1 - - ---I-----t- -- I
I I I I
o 2 4 6 8 x
2. HaHJlHTe nJIOll1ab pOM6a, eCJ1H ero BblCOTa paBH3 5, a Oll"H H3 yrJIOB
paBeH 150°.
3. nepHMerp npHMoyroJlbHHKa paBeH 48, a nJ10111allb paBH3 119. HaJ1-
llHTe 60JIbWYIO CTOPOHY npSlMoyroJlbHHKa.
4. OCHoBaHHSI pasHo6e,npeHHoH TpaneUHH paBHbl 22 Ii 32, a ee nepH-
MeTp paBeH 80. HaUJI.HTe nJlOllla!lb 3TOH TpaneLlHH.
5. nJIOlllallb TpeyrOJIbHHK3 ABC pasHa 27. Ha cTopoHax AC 11 BC
B3HTbi COOTBeTCTBeHHO T04KH M H N r3K, qTO AM : CM BN : CN
7 : 8. HaHJlHTe nJIOlllaJlb TpeyronbHHKa CMN.
BapHaHT 8
1. HaHJlHTe nJlOlllallb qeTbIpexyrOJIbHHKa,
H306pa)KeHHOrO B KoopllHHaTHoH nJIOCKOCTM
H3 )1aHHOM cnpaBa pHcYHKe.
5 ----------
I
4 -------- -T----- I
I I
I I
I I
I I
I I
I
I
2 ---
I
1 --------- I
5
o 2
8 x
2. HaH,llHTe nnOUla.nb pOMoa, eCJIH ero BblCOT3 paBH3 tI3, a OllHH 113
yrJIOB paBeH 1200.

KOHTp0.l1bH8n pa60Ta
83
3. OepHMeTp npHMoyrOJIbHHKa paBeH 42, a nJlOlllaD-b pasHa 104. HaH-
HHTe MeHbwylO CTOPOHY npHMoyroJlbHHK8.
4. OCHoBaHHH paBHo6ellpeHHoH TpaneUMH paBHbI 17 H 41, a ee nepH-
MeTp paBeH 88. HaHJlHTe nJIOlllallb 3TOH TpaneUHH.
5. Ha CTOpOH3X AC, BC TpeyroJlbHHKa ABC B3SITbl COOTBeTCTBeHHO
T04KH M, N TaK, liTO Akf: Cj\J! BN : CN 3 : 2. UJIOlllaJtb Tpe-
yroJlbHHK3 CMN paBHa 18. HaHllHTe nn01ll3Hb TpeyroJlbHHKa ABC.
BapHaHT 9*
1. Ha KJleTlJaTOH 6YMare, HMelOlllett pa3Mep KJIeTOK
5 MM X 5 MM, H306pa)KeH nJlaH ropollcKoro cKBepa,
CM. PHCYHOK (TeppHTopHH CKBepa 06BeJleHa )l{Hp-
HOH J1HHHeM). OnpelteJIHTe MaCWTa6 3Toro H306pa-
>KeHHH, eCJ1H H3BeCTHO, lITO nJIouraJlb CKBepa CO-
CTaBJI5IeT 87 ra (1 ra 10000 M 2 ).
2. nJIOlllallb TpeyrOJIbHHKa ABC paSHa 36. 4eMY MO>KeT 6bITb paBeH
KOCHHYC yrJI3 ABC, eCJIH AB 3V6, BC lO?
3. nepHMerp np5lMoyrOJIbHHKa paBeH 28, a JtJlHH3 J1HarOHaJIH pasHa 11.
HaHJlHTe nJlOllla1!b npHMoyroJlbHHKa.
4. B paBH06eD-peHHOH TpaneUHH ABC D H3BeCTHbi JI.JIHHbI OCHOBaHHH
H 60KOBbiX CTOpOH: BC 6, AD 24, AB CD 41. UYCTb P
T04K3 nepece l leHHS1 ltHarOH3J1eH AC H BD. HaHllHTe nJIOllla,Qb Tpe-
yrOJ1bHHKa BCP.
5. OJIOllla1!b TpeyrOJIbHHKa ABC paBH3 45. Ha cTopoHax AC Ii BC
B3HTbi COOTBeTCTBeHHO TOqKH M H N TaK, liTO AM : CM 4 : 9,
BN : CN 5 : 13. HaHJlMTe nJIOll1aAb TpeyrOJIbHHKa AMN.

84
 6. nnOmaJI.b
.
BapHaHT 1 0*
1. Ha KJIeT4aTOH 6YMare, HMeIOlUeH pa3Mep KJIeTOK
5 MM X 5 MM, H306pa>KeH IIJlaH npo-noBOJlbcTBeHHoro IIIIC:
pblHKa, CM. PHCYHOK (TeppHTopHH pblHK3 06Belle-
Ha >KHpHOH JIHHHeH). OnpeJIeJlHTe MaCWTa6 3Toro
H306pa)l{eHHH, eCJIH H3BeCTHO, 410 nJlOlllan.b PbIH-
Ka paSHa 6,3 ra (1 ra 10000 M 2 ).
2. nJIOmaJ!b TpeyrOJ1bHHKa ABC pasHa 42. 4eMY MO>Ker 6bITb paBeH
TaHreHC yr J13 ABC, eCJIH AB 4V2, BC 15 ?
3. nepHMeTp npSlMoyroJlbHHKa paBeH 34, a JlJIHHa D.HarOHaJlM pasHa 13.
HatiJlHTe nJIOLUBllb npSIMoyrOJIbHHKa.
4. B paBH06eJlpeHHOH rpaneUHH ABC D H3BeCTHbI lLlIHHbl OCHOB3HHH
Ii 60KOBbiX CTOpOH: BC 12, AD 36, AB CD 37. nYCTb P-
T04Ka nepece4eHHH llH3roHaneH AC H BD. 1aHJlHTe nJIOllla)!b Tpe-
yroJlbHHKa ABP.
5. Ha cTopoHax AC H BC TpeyroJlbHHKa ABC B3SlTbi T04KH M H N
TaK, qTO .Al\tf: CM - 2 : 7, BN: CN 8 : 3. n.nolUallb TpeyroJlbHMKa
AMN pasHa 22. HaHllHTe nnOmal1b rpeyr0J1bHHK3 ABC.

YrJ1bJ, BnHCaHHbl€ B OKPY)/(HOCTb
85
 7. OKPY>KHOCTb
7.1. YrJlbl, BnHcaHHble B OKPY}KHOCTb
1. 4epe3 T04KH A, B, C KoopHHHaTHoH nJIOCKOCTH
npOXOJlHT OKPY>KHOCTb c ueHTpOM B TOllKe 0, CM. PH-
CYHOK. HaHJlHTe rpaJIYCHYIO Mepy yrJ1a ABC.
o
2. 4epe3 T04KH A, B, C KoopllHHaTHoH nJIOCKOCTH
npOXOJ1HT OKPY>KHOCTb c ueHTpOM B T04Ke 0, CM. PH-
CYHOK. HaHAHTe rpaJlYCHYIO Mepy yr.na ABC.
o
3. XopJla AB lleJIHT OKPY>KHOCTb H3 HBe .nyrH, rpaJlYCHble BeJlH4HHbi
KOTOpblX OTHOCHTCH KaK 5 : 7. nOD- KaKHM yrJIOM BHJlHa 3Ta Xop!la H3
TOllKH C, npHHa!lJle}KallleH MeHbllleH !lyre OKPY)KHOCTM? OTBeT llaMTe B
rpa.aycax.
4. Xopa AB HeJ1HT OKPY)f(HOCTb Ha JlBe l1yrH, rpaJ!ycbIe BeJlHIIHHbl
KOTOpbIX OTHOCHTCH KaK 4 : 41. non K3KI1M yrJIOM BI111Ha 3T3 xop.na 113
TOlJKH C, npHHal!ne)KallleH MeHbweH Jlyre OKpY)l(HOCTI1? OTBeT ,naHTe B
rpaJlycax.
5. XOPJt3 AB HeJ1HT OKPY)KHOCTb H3 JiBe llyrH, rpallYCHbIe BeJIH4MHbi
KOTOpbIX OTHOCHTCH KaK 11 : 25. nO.Ll KaKHM yrJIOM BHl{Ha 3T3 xopn.a H3
TOlJKH C, npHHallJIe)KallleH 60JIbWeH Jlyre OKPY)KHOCTH? OTBeT JlaHTe B
rpa,nycax.
6. Xoplla AB lleJIHT OKPY)KHOCTb H3 JlBe JlyrH, rpaAYCHbIe BeJlH1.JMHbl
KOTOpbIX OTHOCSlTCSI KaK 7 : 65. nOA KaKHM yrJloM BM)lH3 3T3 xopJla H3
TOqKH C, npHH3LlJIe>KallleH 60JlbWeH llyre OKPY)l{HOCTI1? OTBeT aHTe B
rpa,aycaxe

86
.
7. OKpy>KHOCTb
7. T04Ka 0 - LleHTp onHcaHHOH OKPY>KHOCTH
TpeyrOJIbHHKa ABC, yrJIbl AOB H AOC paBHbI
115 0 H 94° COOTBerCTBeHHO. HaHHHTe rpan.yc-
HYIO Mepy yrJIa A TpeyrOJIbHHKa ABC.
8. TOlJKa 0 - LleHTp onHcaHHOH OKPY)KHOCTH
TpeyrOJ1bHHKa ABC, yrJIbl AOB H AOC paBHbl
131 0 H 143 0 COOTBerCTBeHHO. HaHllHTe rpaltyc-
HYIO Mepy yrJIa A TpeyrOJIbHHK3 ABC.
L
9. TOIIKa 0 - ueHTp onHcaHHOH OKPY)I{HOCTH
TpeyrOJIbHHK8 ABC, yroJl BCO paBeH 44° .
HaHJlHTe rpaJlycHYIO Mepy yrJIa A TpeyrOJIbHH-
Ka ABC.
A
I O. TOlJKa 0 - ueHTp onHcaHHOH OKPY)KHO-
CTH TpeyrOJIbHHKa ABC, yrOJI BCD paBeH 65°.
HaHJlHTe rpaJtycHYIO Mepy yrna A rpeyrOJ1bHHKa
ABC.
A
11. TOIIKH A, B, C, pacnOJIO)l{eHHble Ha OKPY>KHOCTH, .QeJl5IT ee Ha TPH
HenepeCeKalOll.{HeC5I LlyrH: AB, BC HAC, rpaJ!ycHble BeJlHIIHHbI KOTO-
pbIX OTHOCHTCH KaK 2 : 6 : 7. HaCtn.HTe rpaJI.ycHYIO Mepy MeHbWerO 113
yrJIOB TpeyrOJIbHHKa ABC.
12. TOtlKH A, B, C, paCnOJlOiKeHHble Ha OKPY)I(HOCTH, lleJIHT ee 1-18 TPH
HenepeCeKalOlllHeCH llyrH: AB, BC HAC, rpan.ycHble BeJlHlJHHbl KOTO-
pbIX OTHOC5ITCH KaK 3 : 8 : 9. HaHJ!HTe rpaJ!ycHYIO Mepy MeHbwero H3
yr J10B TpeyrOJIbHHKa ABC.
13. TOlfKH A, B, C, pacnOJIO>KeHHble H3 OKPY>KHOCTH, lteJIHT ee Ha TP"
HenepeCeKalOll.{HeCH !lyrH: AB, BC HAC, rpaJlycHble BeJlH4HHbI KOTO-
PbIX OTHOCSlTCSI K3K 3 : 4 : 11. HaHl1HTe rpaJlycHYIO Mepy 60JIblllerO H3
yr.noB Tpeyro.nbHHKa ABC.

,VrJIbJ, BnHCaHHble B OKPY)/(HOCTb
87
14. TOlJKH A, B, C, pacnOJlO>KeHHble Ha OKPY>KHOCTH. ,QeJISlT ee H8 TP"
1.lenepeceK31OIllHecSlllYrH: AB, BG HAC, rpa,nycHble BeJlHlJHHbl KOTO-
pblX OTHOCHTCH KaK 1 : 12 : 23. HaH,QMTe rpanycH}'1O Mepy 60nbUJerO
113 yrJIOB TpeyrOJIbHHK3 ABG.
15. TOqKH A, B, C, D, pacnOJlO)l(eHHbIe H3 OKPY>KHOCTH, HeJlS1T 3TY
OKPY}KHOCTb Ha lJeTblpe HenepeceK810lU H ecH .nyrH: AB, BC, CD "AD,
rpallYCHble BeJlHlJ"Hbl KOTOpblX OTHOCSlTCSI KaK 6 : 7 : 8 : 9 COOTBeT-
CTBeHHO. HaHllliTe rpaJlycHYIO Mepy yr JIa A lJeTblpexyroJlbHHKa ABCD.
16. TOlJKH A, B, C, D, pacnOJlO)l(eHHble Ha o KPY)I(HOCTH, eJlHT 3TY
OKPY>KHOCTb Ha lJeTblpe HenepeCeKalOl.1.lHeCSI nyrH: AB, BC, CD H AD,
rpaJ!ycHbIe BeJlH1.JHHbI KOTOpblX OTHOCSITCH K3K 1 : 3 : 5 : 11 COOTBeT-
CTBeHHO. HaH.QMTe rpanycHYIO Mepy yr JIB B QeTblpexyrOJ1bHHKa ABCD.
17. TOlJKH A, B, C, D, pacnOnO}KeHHble Ha OKPY}KHOCTH, J1eJ1HT 31)'
OKPY)KHOCTb H3 tJeTbIpe HenepeCeK31Ol.1.lHecSinyrH: AB, BC, CD H AD,
rpallYCHble BeJlHqHHbl KOTOpblX OTHOCSITCH KaK 4 : 21 : 9 : 16 COOTBeT-
CTBeHHO. HaHllHTe rpanycHYIO Mepy yr J1a C QeTblpexyrOJ1bHHKa ABCD.
18. TOliKH A, B, C, D, pacnOJlO}KeHHble H3 OKPY>KHOCTH, D.eJIHT 31)'
.
OKPY)KHOCTb Ha 1.JeTblpe HenepeCeKalOlUHeCH nyrH: AB, BC, ,CD MAD,
rpaHycHble BeJlHllHHbl KOTOpbIX OTHOCHTCH KaK 2 : 6 : 7 : 17 COOTBeT-
CTBeHHO. HaHllHTe rpaJ1ycHYIO MepyyrJla D qeTbIpexyroJlbHHKa ABCD.
19. B OKPY}KHOCTH npoBe.neHbl JlHaMeTp AB H nepeceKalOlllaSi 3TOT
AHaMeTp XopJla CD. YroJl ABC paBeH 87°. HaHD.HTe rpanycHYIO Mepy
yrJIa CDB.
20. B OKPY>KHOCTH npoBelleHbl llHaMeTp AB Ii nepeceKalOlll3S1 3TOT
-
JlH3MeTp xoplla CD. YroJl CDB paBeH 22°. HaHJlHTe rpanycHyIO Mepy
yrJla ABC.
21. B OKPY>KHOCTH npOBeJleHbl .n lt3 MeTp AB H
He nepeeKalOI.1.laSl 3TOT llHaMeTp xopn-a CD, "PH
3TOM XOpJlbl AC H B D TalOKe He nepeceK810TCH
(CM. PHCYHOK). YroJI ABC paBeH 17°. HaH)JJiTe
rpaLlycHYIO Mepy yrna GDB.

88
97. OKPY)/(HOCTb
...
22. B OKPY)KHOCTH npOBeJleHbl ,llHaMeTp AB H
He nepeCeKalOI.1.laS1 3TOT D.HaMeTp xoplla CD, npH
3TOM XOp-Ubl AC H B D TalOKe He nepeceK3lOTC51
(CM. PHCYHOK). YroJl CDB paBeH 157 0 . HaHJlH-
Te rpaAycHYIO Mepy yrJ13 ABC.
c
7 .2. lJeTblpexyrOJlbHHK, BnHcaHHblH B OKPY>KHOCTb
23. 4eTbIpexyrOJIbHHK ABCD BnHcaH B OKPY)KHOCTb. 113BecTHO, liTO
LCDB 32° H LABD 53°. HaHllHTe LBPC, fIle P TOqKa ne-
peCeqeHHH JlHarOHaJIeH LJeTblpexyrOJIbHHK3 ABCD.
24. ,LlHarOHaJ1H AC H BD 4eTblpexyrOJIbHHKa ABCD, BnHcaHHoro 8
OKPY>KHOCTb, nepeceKalOTCSI B T04Ke P. rpanycHaH Mepa MeHbweH tlyrH
OKPY/KHOCTH, CTHrHBaeMOH XOPllOH BC, paSHa 1100, a rpaD-ycHaH Me-
pa MeHbweH !lyrH oKpymHocTH, cTHrHBaeMoM XOPJlOH AD, pasHa 155°.
Hal1HHTe rpan.ycHYIO Mepy yrJI3 APD.
25. 4eTbipexyroJlbHHK ABC D BnHcaH B OKPY)KHOCTb. 113BecTHO, liTO
LCDB 41° H LACD 56°. HaHJlHTe LBQC, rlle Q - TOtJKa ne-
peCeqeHHSI npOAOJI>KeHHH CTOpOH AB Ii CD.
26. npOllOJl/KeHHSI CTOpOH AB H CD lleTbIpexyrOJ1bHHKa ABCD,
BnHcaHHoro B OKPY)I{HOCTb, nepeceKalOTcH B TOllKe Q. rpaJ!ycHaH Me-
pa MeHbUleH .nyrH OKPY>KHOCTH, CTHrHBaeMOH XOPllOH BC, pasHa 74 0 , a
rpal{ycH3ff Mepa MeHbllleH JlyrH oKpymHOCTI1, CTSIrHBaeMOH XOPllOH AD,
pasHa 147 0 . HaHLlHTe rpaJ!ycHYIO Mepy yrJIa AQD.
27. llBa yrJIa qeTblpexyrOJlbHHKa, BnHcaHHoro B OKPY)KHOCTb, paBHbl
61 0 H 83°. HaH,QHTe rpallycHYIO Mepy 60JIblllerO 113 OCTaBllJHXCH yrJloB
..
3Toro tJeTblpexyrOJIbHHKa.
28. ,LlBa yrJla qeTblpexyrOJlbHHKa, BnHcaHHoro B OKPY>KHOCTb, paBHbI
77° H 99°. HaHJlHTe rpallycHYIO Mepy MeHblllero H3 OCT3BlliHXCH yrJloB
..
3Toro qeTblpexyrOJIbHHKa.
29. YrJlbI A,B,C 4eTblpexyroJlbHHK3 ABCD OTHOCHTCSI KaK 3: 11 : 12

lJeTbJpexyrO.llbHHK, BnHCaHHblH 8 OKpy>KHOCTb
89
COOTBeTCTBeHHO. HaHJlHTe rpaJlYCHYlO Mepy yrJla D, eCnH OKOJlO llaH-
..
HOrO LJeTblpexyrOJIbHHKa MO)KHO OnHC8Tb OKPY)I{HOCTb.
30. YrJlbl A, C 11 D QeTbIpexyrOJIbHHK3 ABCD OTHOCHTCH KaK 5: 7: 8
COOTBeTCTBeHHO. HattUMTe rpaJlycHYIO Mepy yrJl3 B, eCJIH OKOJIO llaHHO-
..
ra LJeTbIpexyrOJIbHHK3 MO)KHO onHcaTb OKPY)I{HOCTb.
31. 4eTbIpexyroJlbHHK ABC D snHcaH B OKPY)I{HOCTb. 113BecTHo, QTO
yrJ1bI A H B OTHOCHTCSI KaK 3 : 2, a yrJIbl C H D OTHOCHTCH KaK 3 : 8.
HaHJlHTe rpaJlycHYIO Mepy 60JIblllerO 113 yrJloB 3Toro qeTblpexyrOJIbHJ.i-
Ka.
32. 4eTblpexyr0J1bHHK ABC D BnHcaH B OKPY>KHOCTb. 113BecTHo, 4TO
yrJIbI A H B OTHOC5ITCH KaK 6 : 5, a yrJlbl C H D OTHOCSlTCSI KaK 3 : 5.
HaHJlHTe rpanycHYIO Mepy MeHbwero H3 yrJIOB 3Toro lJeTblpexyrOJIbHH-
K8.
33. 4eTbipexyroJlbHHK ABCD BnHC8H B OKPY)I{HOCTb. YroJI ABC pa-
BeH 115 0 , yroJi ADB paBeH 56°. HaHD.liTe rpa-aycHYIO MepyyrJIa BAC.
34. 4eTbipexyroJlbHHK ABCD BnHC8H B OKPY>KHOCTb. YroJl ABC pa-
BeH 133 0 , yroJl AD B paBeH 33°. HaHJlHTe rpallycHYIO Mepy yr JI3 BAC.
35. 4eTbipexyroJlbHHK ABCD BnHcaH B OKPY)KHOCTb. YroJl ABC pa-
BeH 170 0 , yro.n DAC paBeH 89°. HaHllHTe rpaJlyclfYIO MepyyrJla ACD.
36. 4eTblpexyroJlbHHK ABCD BnHcaH B OKPY)I{HOCTb. YroJl ABC pa-
BeH 178 0 , yroJl DAC paBeH 81 0 . HaHLlHTe rpaDYcHYIO Mepy yr Jl8 AC D.
37. 4eTblpexyr0J1bHHK ABCD BnMC3H B OKp)')KHOCTb, P TOlJKa ne-
peCelJeHHSI ero .QHarOH3J1eH. J{OK3)KHTe, liTO AP · CP BP · DP.
38. 4eTbIpexyr0J1bHHK ABCD BnHcaH B OKp)')KHOCTb, Q TOlJKa ne-
peCeqeHHSI npOllOJl)KeHHH CTOpOH AB H CD. J{OKa)l{MTe, liTO
AQ · BQ CQ · DQ.
39. 4eTblpexyrOJIbHHK ABCD BnHC3H B OKPY)I{HOCTb, P
peCelJeHHH ero JlHarOHaJIeH. Vl3BeCTHO, liTO AP 10, BP
CP 15. HaH.nHTe Jt.nHHY JlHarOHaJlH BD.
T04Ka ne-
12,

90
7. OKPY)/{HOCTb
'"
40. 4eTbipexyroJlbHHK ABCD BnHCaH 8 OKPY)I{HOCTb, P TOtlKa ne-
peCetteHHSI ero llHarOHa.neH. 113BecTHo, tlTO AP 20, BP 9,
BD 27. HaHllHTe DJlHHY J1HarOH3JIH AC.
41. 4eTbipexyrOJIbHHK ABCD BnHcaH B OKPY>KHOCTb, Q - TOtIKa ne-
peCeqeHHSI npon.OJl}l{eHHH CTOpOH AB H CD. 113BecTHo, lITO AB 18,
AQ 28, DQ 32. HaHJ!HTe LlJIHHY CTOpOHbI CD.
42. 4eTbipexyrOJIbHHK ABCD BnHC3H B OKPY)KHOCTb, Q T04Ka ne-
peCeqeHHSI npo,noJ)}I{eHHH CTOpOH AB H CD. 113BecTHo, liTO AB 24,
AQ 56, CQ 28. HaHllJiTe LlJIHHY CTOpOHbI CD.
43. 4eTbIpexyrOJIbHHK ABCD BnHcaH B OKPY)I{HOCTb. Vl3secTHo, IJTO
AB 2, CD 6, AC 9, BD 11. HaMJlHTe LlJIHHY oTpe3Ka AP,
rIle P TOQK3 nepeCelJeHHH J1HarOHaJ1eH AC H B D.
44. 4eTbipexyroJlbHHK ABCD BnHcaH B OKPY)KHOCTb. Vl3BeCTHO, lITO
AB 3, CD 7, AC 14, BD 16. HaHllHTe JlJIHHY oTpe3Ka AP,
rlle P - TOIlKa nepeCeqeHHH l1HarOHaJleH AC Ii B D.
45.* 4eTbipexyroJlbHHK ABCD BnHcaH B OKpy>KHOCTb. Vl3seCTHO, liTO
AB 5, AC 6, BD CD 10. HaCtUI1Te J1JIHHY oTpe3Ka AQ, rIle
Q TOllKa nepeCeQeHHS1 npOAOJl>KeHHH CTOpOH AB H CD.
46.* 4eTblpexyrOJIbHHK ABCD BnHcaH B OKPY>KHOCTb. H3BeCTHo, liTO
AB 14, AC 15, BD 18, CD 19. HaHLlHTe JlJ1HHY oTpe3Ka
AQ, rlle Q - TOlJKa nepeCelJeHHH npOllOJI>KeHHH CTOpOH AB H CD.
7 .3. nepneH)l.HKYJlHp 03 QeHTpa OKPY)I(HOCTH K xOPAe
47. 1!J1HHa XOpltbI OKPY}I{HOCTH paBH3 48, a paccToHHHe OT ueHTpa
OKPY}KHOCTH )10 3TOH XOpJlbI pasHo 32. HaHAHTe llHaMeTp OKPY}I{HOCTH.
48. ,UJIMHa XOp-U bI OKPY>KHOCTH paBH3 70, a paccToS1HHe OT ueHTpa
OKPY}I{HOCTH JlO 3TOH XOpn.bI pasHo 12. HatiDJ"Te J!HaMeTp OKPY>KHOCTH.
49. B OKPY>KHOCTH, JlH3MeTp KOTOpOH paBeH 100, npOBeJleHa xopna,
u
TaK3S1, liTO paCCTOS1HHe OT ueHTpa OKPY>KHOCTH ,no 3TOH XOPD.bl paBHo
14. HaHJlHTe MHHY 3TOH XOPlt bI .

nepneHJJ.HKy.nnp H3 ueHTpa OKPY)/(HOCTH K XOpJI.e
91
50. B OKPY)I{HOCTH, JlHaMeTp KOTOpOH paBeH 90, npOBeJleHa xoplla, T3-
KaH, tJTO paccToHHHe OT LleHTpa OKPY)KHOCTH no 3TOH XOpJlbI pasHo 27.
HaHllHTe 11J1HHY 3TOH XOpLlbl.
51. B OKPY>KHOCTH C llH3MeTpoM, paBHblM 70, npoBe,neHa xoplla J!JIH-
Ha KOTOPO" paBHa 56. HatiJlHTe paccToHHHe OT LleHTpa OKPY>KHOCTH 110
u
3TOH XOPJlbI.
52. B OKPY>KHOCTH C JIHaMeTpoM, paBHblM 78, npo Be lleH3 xop.aa, JlJIH-
Ha KOTOpOH pasHa 72. HaHJJ.HTe paccToHHHe OT ueHTpa OKp}')KHOCTH ,no
u
3TOH XOpllbI.
53. OTpe3KH AB 11 CD HBJl5IIOTCH xop.aaMH OKPY)KHOCTI1. HaHllHTe
paCCT05lHHe OT ueHTpa OKPY>KHOCTH ,no XOpllbl CD, eCJIH AB 80,
CD 18, a paccToHHHe OT ueHTpa OKPY>KHOCTH !to XOpHbI AB paBHo 9.
54. OTpe3KH .4B Ii CD 51BJI5IIOTC5I XOpJlaMH OKPY)I{HOCTH. HaH,llHTe
paccToHHHe OT ueHTpa OKPY>KHOCTI1 .no XOpLlbl CD, eCJ1H AB 30,
CD 48, a paccToHHHe OT ueHTpa OKPY)I{HOCTH llO XOpJlbl AB paBHo 20.
55. OTpe30K AB HBJlSleTCH XOPJlOH OKPYiKHOCTH C ueHTpoM B TOlJKe O.
TOllKa P, Jle>KaLl.l851 H3 oTpe3Ke AB, TaKOB3, LITO AP 6, OP 16,
BP 24. HaHHTe pa,llHYc OKPY)KHOCTH.
56. OTpe30K AB HBJIS1eTCH XOPJlOH OKPY*HOCTH C LleHTpOM B TOllKe o.
TOllKa P, J1e>KaIll351 H3 OTpe3Ke AB, raKOBa, liTO AP 6, OP 13,
BP 20. HaHllHTe paDJiYc OKPY)I{HOCTH.
57. B OKPY>KHOCTH npOBeJIH XOp.abl AB H CD, KOTopble nepneHllM-
KYJ1HpHbl Jtpyr llPYry H nepeceKalOTcSI B TOqKe P T3K, lfTO AP 15,
BP 35, OP 21, DP 25. HaH!lHTe paJtuyc OKPY>KHOCTH.
58. B OKPY>KHOCTH npOBeJIH XOpAbI AB H CD, KOTopbIe nepneHD-H-
KYJlHpl-lbl Jtpyr Jtpyry Ii nepeceKalOTCH 8 T04Ke P TaK, liTO AP 30,
B P 24, C P 20, D P 36. HaHllHTe paccToHHHe OT T01.IKH P
ll-O ueHTpa OKPY>KHOCTH.
59. B OKPY)KHOCTb pa.nHyca 13 BnHcaHa TpaneUHSI, OCHOBaHHSI KOTO-
POH paBHbl 10 H 24, npHlJeM UeHTP OKPY)KHOCTH J1e>KHT BHYTpH Tpane-

92
9 7 . OKPY)/(HOCTb
..
Ll"H. HaH,QHTe BblCOTY 3TOM TpaneUHH.
60. 8 OKPY)KHOCTb paHyca 20 BnHcaH3 TpaneUHSI, OCHOBaHHH KOTO-
pOH paBHbI 24 H 32, npHlJeM ueHTp OKPY)KHOCTH ne>KHT BHe TpaneLlHH.
· HaHLLHTe BbIcory 3TOH TpaneUHH.
.
61.* B OKPY/KHOCTb pa!lHyca 7 BnHcaHa TpaileUHSI ABCD C OCHOB3-
HHHMH AD Ii BO. HatiD-HTe BblCOTY TpaneUI1H, eCJ1H yrOJl ABD paBeH
60t"), a KOCHHYC yr J1a BAG paBeH 0,8.
62.* 8 OKPY>KHOCTb pan.Hyca 9 BnHcaHa TpaneUHH ABCD C OCHOBa-
HH5IMH AD H BG. H3SeCTI-IO, 4TO yrOJ1 ABD paBeH 30°, a CHHYC yrJIa
BAG paBeH 0, 7. HaHllHTe MHHY cpellHeH JlHHHH TpaneUHH.
7 .4. KaCaTeJlbHaH K OKPY>KHOCTH
63. HaHLlHTe rpaJlYCHYIO Mepy yrJIa AGO, eCJIH J1Y4 GA KacaeTCSI
OKPY>KHOCTH C LleHTpoM B TOlJKe 0 (CM. PHCYHOK), a rpallycHaH Me-
pa MeHbweti JlyrH 3TOH OKp)'e)KHOCTH, 3aKJIIO l leHHOH BHYTP" yrJl3 AGO,
pasHa 63°.
A
c
64. HaHHTe rpaJlYCHYIO Mepy yr.na AGO, eCJIH JIY(f OA KacaeTCH
OKPY>KHOCTH c ueHTpoM B T04Ke 0 (CM. PHCYHOK K npeD-blltYllleH 3a,l1a-
lie), a rpanycHaH Mepa 60JIbWeH llyrH 3TOH OKPY}KHOCTH, 3aKJ1IOQeHHOH
BHYTPH yrJ1a AGO, paSHa 118°.
65. J1Y4 0 A KacaeTCH OKPY>KHOCTH c ueHTpOM B TOlJKe 0 ( CM. PHCYHOK
K 33J{atJe N263). HaHAHTe rpan.YCHYIO Mepy MeHbweH .LlyrH OKPY}KHOCTH,
3aKJIIOlJeHHOH BHYTPH yrJl3 AGO, eCJIH LAGO 25°.
66. .flyq G A KacaeTCH OKPY>KHOCTH C LleHTpOM B TOI.JKe 0 (CM. PHCYHOK
K 3aJt8lJe N263). HaHllHTe rpaltycHYIO Mepy 60JIbWeH .nyrH OKPY}KHOCTH,
3aKJIIOlleHHOH BHYTPH yrJIa AGO, eCJJM LAGO 26°.
67. ,UaHa OKPYJKHOCTb H T04K3 A, J1e>K3111351 BHe 3TOH OKPY>KHOCTH. 113

KaCareJ1bH8H K OKPY}/(HOCTH
93
T04KH A npoBelleHbl D.Be npSlMble, 011H3 113 KOTOpblX KaCaeTCSI .a3HHOH
OKPY>KHOCTH B TOlJKe L, a n.pyraH nepeceKaer OKPY>KHOCTb B T04Kax B
H C. ,UOKa>KHTe, liTO AL 2 AB · AC (KBadpam KacameAbHOU pa8eH
npOU3BedeHUIO ompe3K08 ceKyU{eu).
68. JIYLJ AB KacaeTCS1 OKPY>KHOCTH c ueHTpOM B TOtlKe 0, a npSlM3S1
AD nepeceKaeT 3TY OKPY>KHOCTb B T01JKaX P H Q TaK, KaK nOKa3aHO
Ha JlaHHOM HHme pHcYHKe.
B
HaHllHTe paJlJtyc OKPY>KHOCTH, eCJ1H H3BeCTHO, QTO:
a) AB 21, AP 9; 6) AB 18, AP 12;
B) AB 40, AQ 50; r) AB 12, AQ 16.
69. ,UaH3 OKPY)I{HOCTb c UeHTpOM B TOlJKe D pall.Hyca 15 H TOlJKa A,
paCCTOHHHe OT KOTOPOH 110 T01JKH 0 paBHo 25. 113 T01.JKH A npoBeJte-
HbI JIBe IipHMbIe, KacalOLllHeCH llaHHOH OKPY»<HOCTH B TOIIKax M H N.
HaHllHTe JL1lHHY OTpe3Ka MN.
.
70. llaH3 OKPY}KHOCTb c ueHTpoM B T04Ke 0 paJlHyca 18 Ii T01.JKa A,
paCCTOSlHHe OT KOTOPOH .no T04KH 0 pasHo 30. 113 TOlJKH A npoBe.ne-
HbI JlBe npSIMbIe, KBcalOLllHeCH Jl3HHOH OKPY}KHOCTH B TOlJKaX M H N.
HaHJ!MTe ,llJIHHY oTpe3Ka MN.
71.* .aaHa OKPY}KHOCTb c ueHTpoM B TOIJKe 0 H TotJKa A, Jle}KaLU3H
SHe 3TOH OKPY}KHOCTM. 113 TOllKH A npOBeJ!eHbI ,l1Be npHMble, KacalO-
LllHecS1 AaHHoH OKPYIKHOCTH B TOIJK3X !vI H N. HaHJ1HTe paJlHYc JlaHHOH
OKPYJKHOCTH, eCJ1H AO 40, MN 38,4 H llOnOJIHHTeJlbHO H3BeCTHO,
lfToAM <OM.
72.* ,UaHa OKPY)I{HOCTb c ueHTpoM B TOIIKe 0 H TOIIK3 A, JIe}KawaH
BHe 3TOH OKPY}KHOCTH. 113 T01JKH A npOBeJleHbl JtBe npstMble, KaC31O-
LllHecSI llaHHoH OKPY>KHOCTH B TOqKaX M H N. HaHJ{HTe pa,lluyc l(aHHOH
OKPY)KHOCTH, eCJlH AO 60, MN 57,6 11 JlOnOJlHHTeJlbHO H3seCTHO,
liTO AM > OM.

94
97. OKPY)J(HOCTb
..
73. B TpeyrOJIbHHKe ABC yrOJJ C paBeH 45°, AB
pallHYc onHcaHHOH OKPY)I{HOCTH TpeyroJlbHHKa ABC.
7 .5. BnHcaHHaH H onHcaHHaSi OKPY>KHOCTH T p e yro JlbHHKa
74. B rpeyrOJIbHHKe ABC yroJl C paseH 30°, AB
pallHYc onHcaHHoH OKPY)KHOCTH TpeyrOJIbHHKa ABC.
75. B Tpeyr0J1bl1HKe ABC yroJl C paseH 60°. PallHYc onHC3HHOH
OKPY}I{HOCTH TpeyrOJ1bHHKa ABC paBeH 2 J3. HaHllHTe CTOPOHY AB.
30.. H a HllHTe
76. B TpeyrOJIbHHKe ABC yroJ1 C paBeH 45°. PaJlHYc onHc3HHoH
OKPY>KHOCTH TpeyrOJIbHHK3 ABC paBeH 31 J2. l-IaHJ{HTe CTOPOHY AB.
77. TOtIKa G - LleHTp BnHcaHHOH OKPY>KHOCTM
TpeyroJlbHHKa ABC, yr JIbl GAC H OBC paBHbI
24° H 26° COOTBeTCTBeHHO. HaHJlHTe rpallycHYIO
Mepy yr Jl3 C TpeyroJlbHHKa ABC.
c
A
78. TOllKa 0 - ueHTp BnHcaHHOH OKPY>KHOCTH
TpeyroJlbHHKa ABC, yrJ1bl GAC H OBC paBHbl
21 0 H 34° COOTBeTCTBeHHO. HaHHHTe rpallycHYIO
Mepy yrJIa C TpeyroJlbHHKa ABC.
c
A
79. DOKoBble CTOpOHbl paBH06eJIpeHHOro TpeyroJlbHHKa paBHbl 26, a
OCHOBaHHe pasHo 48. HaHJtHTe paAHYc onHcaHHoH OKPY>KHOCTH 3Toro
TpeyrOJIbl-lHK3.
80. DOKoBbIe CTOpOHbI paBH06eJlpeHHOro TpeyrOJIbHHKa paBHbl 34, a
OCHOB3HHe paBHo 60. HaHJlHTe pa)lHYc onHcaHHOH OKPY>KHOCTH 3Toro
TpeyrOJIbHHK3.
81.* DOKoBbIe CTOpOHbI paBHo6eLlpeHHoro TpeyrOJIbHHKa paBHbl 80, a
pa,QHYc Onl1CaHHOH OKPY>KHOCTH paBeH 50. HaHLlHTe MHHY OCHOBaHHS1
3Toro TpeyroJlbHHKa.
82_* DOKoBbIe CTOPOHbI paBH06eJlpeHHOro TpeyroJlbHHK3 paBHbI 15, a

BnHcaHHan H OnHC8HHaH OKPY)/(HOCTH rpeyroJlbHHKa
95
paJlHYC OnHCaHHOH OKPY/KHOCTM paBeH 12,5. HaHllHTe JJ.JIHHY OCHOB8-
HHSI 3Toro TpeyrOJIbHHKa.
83. 6oKoBbie CTOpOHbl paBHo6ellpeHHoro TpeyrOJIbHHK3 paBHbl 17, a
OCHOBaHHe paBHo 16. HaHJlHTe pallHYc BnHcaHHoH OKPY>KHOCTH Tpe-
yroJlbHHK3.
84. 6oKoBbIe CTOpOHbl paBHooellpeHHoro TpeyrOJIbHHK3 paBHbl 41, a
OCHOBaHHe paBHo 18. HaH,[HTe paD.HYc BnHcaHHoH OKPY}KHOCTH Tpe-
yrOJIbHHKa.
85. B npHMoyroJlbHoM TpeyroJlbHMKe a
Ii b - J]JIHHbl KaTeTOB, C ,llJIHHa rHno-
TeHY3bl. lloK3>KHTe, liTO paJlHYC OKPY>K-
u
HOCT", KacalOllleHCSI rHnOTeHY3bl H npo-
LlOJI>KeHHH KaTeTOB 3Tor o TpeyroJlbH HKa
.
86. B npSlMoyrOJIbHOM TpeyroJlbHHKe a
11 b ,llJIMHbl KaTeTOB, C lIJIHHa rHnOTe-
HY3bl, CM. PHCYHOK. nOKa>KHTe, qTO paD.HYC
u
BnHcaHHOH OK P Y >KHOCTH 3 Toro Tpeyronb-
c
.
87. rHnoTeHY3a npSlMoyrOJ1bHOrO TpeyrOJIbHHKa paBHa 25, a paD-HYc
BnHcaHHoH OKPY}KHOCTH paBeH 5. HaHJlHTe nepHMerp 3Toro TpeyrOJIb-
HHKa.
88. rHnoTeHY33 npHMoyroJlbHoro TpeyrOJIbHHKa paBHa 39, a pallHYc
BnHcaHHOH OKPY>KHOCTH paseH 6. HaHl!MTe nepHMeTp 3Toro TpeyrOJIb-
HHKa.
89. PaJlHYc OKPY}KHOCTH, BnHcaHHOH B npSlMoyroJlbHblH TpeyrOJlbHHK,
paBeH 4. HaHJtHTe rHnoTeHY3Y, eCJIH TOqKa K3C3HMSI C BnHcaHHoH OKPY}K-
HOCTblO J!eJIMT ee Ha oTpe3KM, lIJIHHbl KOTOPbIX OTHOCSlTCSI KaK 2 : 3.

96
7. O«PY»<HOCTb
90. Pa,llHYc OKPY)KHOCTH, BnHcaHHOH B npS1MoyroJlbHblH TpeyrOJlbHMK,
paBeH 8. HaiillHTe rHnoTeHY3Y, eCJlH TOqK3 KacaHHH C BnHcaHHOH OKPY)I{-
HOCTblO lleJIHT ee Ha oTpe3KH, JlJIHHbl KOTOpblX OTHOCflTC5I KaK 3 : 10.
91. K OKPYlKHOCTH, BnHcaHHOH B rpeyroJlbHHK
ABC, npOBe1!eHa KaCaTeJlbHaH, nepeceKalOllla5I
CTOpOHbI AC Ii BC B TOIIKax P H Q COOTBeT-
CTBeHHO, CM. PHCYHOK. 113secTHO, liTO AB 567,
a nepHMeTp TpeyrOJIbHHKa ABC paBeH 2024.
HaHLlHTe nepHMeTp TpeyroJlbHHKa C PQ.
A
92. K OKPY>KHOCTH, BnHcaHHOH B TpeyrOJIbHHK
ABC, npoBeJteHa KaCaTeJlbHaS1, nepeceK3101113SI
CTOpOHbl AC H BC B TOlfKax P H Q COOTBeT-
CTBeHHO, CM. PHCYHOK. Vl3BeCTHO, LITO nepHMeTp
TpeyroJlbHHK3 ABC paneH 1062, a nepHMeTp
TpeyroJlbHHKa C PQ paBeH 678. HaHHTe AB.
A
93. K oKpyml-lOCTH, BnHcaHHOH B TpeyrOJIbHHK
ABC, npoBelleHa KaCaTeJlbHaH, napaJ1JIeJ1bI13S1
cTopoHe AB H nepeceKalOlllaH CTOpOHbl AC, BC
B TOlJK8X P 11 Q COOTBeTCTBeHHO. Y13secTHO, liTO
AB 15, a nepHMeTp TpeyrOJ1bHHKa C PQ paBeH
20. HaH.QHTe llJIHHY orpe3Ka PQ.
c
A
94. K OKPY>KHOCTM, BnHC8HHOH B TpeyroJlbHI1K
ABC, npoBe,neHa KaC3TeJlbHaH, napaJIJIeJlbHaH
cTopoHe AB H nepeceKalOIlla51 CTOpOl-lbl AC, BC
B TOIIKaX P H Q COOTBeTCTBeHHO. Y13BecTHo, liTO
AB 16, a nepHMeTp rpeyroJlbHHK3 ABC paBeH
80. HaH,UHTe llJIHHY oTpe3K3 PQ.
c
A

IJ/IHcaHHan H OnHCaHHaR OKJJY»<HOCTH MHoroyrOJ/bHHKOB
97
7.6. BnHcaHHaH H onHcaHHaH OKPY)I(HOCTH
MHoroyroJlbHHKOB
95. OKOJIO OKPY>KHOCTH, pan.MYC KOTOpOH paseH 7, onHcaH MHoro-
yrOJlbHHK, nepHMeTp KOToporo paBeH 64. HaHllHTe nJJOllla,nb 3Toro MHO-
royroJlbHHKa.
96. OKOJIO OKPYiKHOCTH, paJlHYc KOTOpOH paBeH 5, onHC3H MHoro-
yrOJIbHHK, nepHMeTp KOToporo paBeH 34. HaHltHTe nJIOlllallb 3Toro MHO-
royroJlbHHKa.
97. OKOJlO OKPY>KHOCTH, pallHYc KOTOpOH paReH 11, onHC3H MHoro-
yrOJ1bHHK, nJ10lllaLlb KOToporo paBHa 396. HaMJlHTe nepHMeTp 3Toro
MHoroyrOJIbHHKa.
98. OKOJlO OKPY/KHOCTH, paJlHYC KOTOpOH paBeH 13, onHcaH MHoro-
yrOJlbHI1K, nnOlllan.b KOToporo pasHa 2015. HaHJI.HTe nepHMeTp 3Toro
MHoroyroJlbHHKa.
99. OK0J10 OKPYJKHOCTH onHcaH MHoroyrOJIbHHK, nJlOm311b KOToporo
paBI-Ia 95, a nepHMeTp paBeH 50. HaHllHTe pallHYc 3TOH OKPY>KHOCTH.
1 00. OK0J10 OKPY>KHOCTM onHcali MHoroyrOJIbHHK, nJlOllla,llb KOToporo
paBHa 143, a nepHMeTp paBeH 110. HaHltHTe paHYC 310M OKPY>KHOCTH.
1 0 I. CTopoHa pOM6a paBHa 30, a OCTpbIH yro.n pOM6a paBeH 30°. Haii-
)!HTe paJlHYc BnHC8HHOH OKPY>KHOCTH pOM6a.
I 02. OCTpblH yroJl pOM6a paBeH 45°, a pa.QHYc BnHc3HHoH OKPYJKHO-
CTH 3Toro pOM6a paBeH 504V2. HaHJlHTe CTOPOHY pOM6a.
1 03. PaJ1HYc onHcaHHOH OKPY>KHOCTH npaBHJlbHOrO lllecTHyroJlbHHKa
paBeH 3J3. Hai1-UHTe palIJ1Yc OKPYJKHOCTH, BnHcaHHOH B 3TOT weCTH-
yrOJ1bHHK.
1 04. PanJ-tyc BnHcaHHOH OKPY>KHOCTH npaBHJlbHOrO wecTHyroJlbHHK3
paBeH 6J3. HaH.QHTe paJlHYc OKPY>KHOCTH, onHcaHHOH BOKpyr 3Toro
lllecTHyrOJI bHHK3.
I ICOMCTP- 7_8 kn. nOl1f K OIJ TCMo TCCTbl

98
7. OKPY)/(HOCTb
1 05. CTopoHa npaBtlJlbHOrO nSlTHyrOJIbHHKa paSHa 37 Sil136°. HaHJ1H-
u
Te pan.HYC onHcaHHOH OKPY)KHOCTH 3Toro n5lTHyroJlbHHKa.
1 06. CTOpOH3 npaBHJlbHOrO n.eBHTHyro.nbHHK3 paSHa 10 sin 20°. Hati-
u
)lHTe paJ!HYc onHcaHHOH OKPYJKHOCTH 3Toro nHTHyrOJIbHHK3.
I 07. naH npaBHJlbHbIH 24-yronbHHK. HaHJlHTe rpallycHYIO Mepy yrJ13
u u
MeiKJlY era CTOpOHOIi H paJlMycoM onHcaHHOH OKPY>KHOCTH.
1 08. B npaBHJlbHOM N -yrOJ1bHHKe yrOJ1 Me)KllY CTOpOHOH H pallHYCOM
u .. u
onHcaHHOH OKPY)I{HOCTH, npOBeJleHHblM B OllHY H3 BeplllHH 3TOH CTOpO-
Hbl, paBeH 84°. HaHJlHTe N.
1 09. llaH npaBHJ1bHblH 45-yr0J1bHHK. HaHllHTe rpanYCHYIO Mepy yr-
U ..
.na Me)Kn.y ltBYMH paJlHycaMH BnHCaHHOI-f OKPY>KHOCTH, npOBen.eHHbIMH B
TOllKH KacaHHSI C .LI.BYMH COCe1!HHMH CTopOHaMH.
11 O. B npaBHJlbHOM N -yro.nbHHKe yroJ1 Me>KJlY llBYM5I pallHyc3MH BnH-
u ..
caHHOH OKPY}I(HOCTH, npoBelleHHblMH B TOqKH KaCaHH5I C D.BYMSI coceJ!-
HHMH CTOpOH3MH, paBeH 6°. HaHllHTe N.
111. llaH npaBHJIbHblH 18-yroJlbH"IK. HatiD.HTe rpallycHYIO Mepy yrJIa
" ..
MelKn.y paAHycoM BnHcaHHOH OKPY}KHOCTH, npOIJeJleHHblM B T04KY Ka-
C3HHSI co CTOpOHOH AB, H pallHycoM onHC3HHOH OKPYiKHOCTH, 11poBe-
JleHHbIM B BepwHHY A.
112. B npaBHJ1bHOM N -yrOJIbHHKe yroJ1 Me)KllY paJlHycoM BnHcaHHOH
OKPY*HOCTH, npOBeJleHHbIM B TOllKY KacaHHH co CTOpOHOH AB, H pa-
JIHYCOM onMC3HHOH OKPY)I{HOCTH, npoBelleHHbIM B BepWHHY A, paBeH
2,5 0 . HaH.nHTe N.
7.7. LJeTblpexyrOJlbHHK, 8 KOTOpblH BnHcaHa OKPY)l{HOCTb
113. OKono OKPY)I{HOCTH onHC3H3 TpaneUHH, LlJIHH3 CpeJlHeH J1I1HHH
KOTOPOH paSHa 0,25. HaHJ{HTe nepHMeTp TpalleUHl1e

LleTblpexyrO.RbHIIK, B KOTOpblii BnHCaHa OKPY>KHOCTb
99
1 14. OKono OKPY)KHOCTH OllHcaHa TpaneUHSI, nepHMeTp KOTOpOH pa-
BeH 114. HaHJlHTe nJlHHY cpellHeH JIHHHH 3TOH Tpaneu"".
115. nepHMeTp npHMoyrOJIbHOH TpaneLlHH, onHC311HOH OKOJlO OKPY>K-
l"iOCTH, paBeH 116, a 60JIbWaS1 113 60KOBbiX CTOpOH paBHa 36. HaHltHTe
paJlHYc OKPY)KHOCTM.
116. CpeJtHSlSI JIHHHH npS1MoyroJlbHoH TpaneUHH, onHC3HHOH OKO.nO
OKPY)KHOCTH, paSHa 61, a 60J1bWaH H3 60KOBbIX CTOpOH paBHa 87. Hai1-
nHTC pallHYc OKPY)KHOCTli.
117. nepHMeTp tleTblpexyrO.flbHHKa paBeH 140, O,QHa "3 ero CTOpOH
paBHa 35, a llpyraH 34. HaHllHTe 60JIbWYlO 113 OCT3BUJHXCH CTOpOH
..
3Toro lleTbIpexyrOJlbHHKa, eCJI1 H3BeCTliO, lITO B Hero MO)KHO BnHC3Tb
OKPY>KHOCTb.
118. nepHMeTp 4eTbipexyrOJIbHHKa paBeH 58, onHa H3 ero CTOpOH pas-
Ha 15, a llpyraH - 16. l-JaHLlHTe MeHbwylO H3 OCTaBlllHXCH CTOpOl-l 3Toro
..
lleTblpexyrOJlbHHK3, eCJ1H 113BeCTHO, 4TO B Hero MO)KHO BnHC3Tb OKPY)K-
HOCTb.
119. 113BecTHo, 4TO B 4eTblpexyroJlbHHK BnHcaHa OKPY>KHOCTb, a JlJlH-
I-Ibl Tpex 113 CTOpOH 3Toro lleTbIpexyrOJIbHHK3 paBHbl 16, 17, 20. KaKoe
HaH60JlbWee 3Ha t leHHe MOiKeT HMeTb llJ1HHa IleTBcpToH CTOpOlfbl 3Toro
lieTbipexyroJlbHHK3 ?
120. 113BecTHo, LITO B 4eTbIpexyrOJIbHHK BnHcaHa OKPY)KHOCTb, a 11.11"-
Hbl Tpex H3 CTOpOH 3Toro tleTblpexyroJlbHHKa paBHbl 90, 99, 101. KaKoe
.. u
H3HMeHbwee 3HaqeHHe MO}l{eT HMeTb 1I.J]HHa lJeTBepToH CTOpOHbI 3Toro
tieTblpexyrOJIbHHK3 ?
121.1{JIH CTOpOH lleTbIpexyr0J1bHHK8 ABCD, onHcaHHoro OKono OKPY-
IKHOCTH, BblnOJlHeHbl COOTHOWeHH5I: AB : BC 4 : 5, AD : CD 8 : 9.
nepMMeTp lleTblpexyroJlbHHKa ABCD pase" 104. HaitllHTe MeHbwylO
..
H3 CTOpOH 3Toro lleTblpexyrOJIbHHKa.
122.1{Jl5) CTOPOH lJeTblpexyrOJIbHHKa ABCD, onHcaHHoro OKono OKPY-

100
7. OKPYJKHOCTb
iKHOCTH, BbInOJlHeHbl COOTHOWeHHH: AB : BC 3 : 7, AD : CD 5 : 13.
nepHMeTp 4eTblpexyr0J1bHHKa ABCD paBeH 532. HaH!lHTe 60JIbWYIO
..
113 CTOpOH 3TOrO lleTblpexyrOJIbHHKa.
7 .8. KacaHHe ABYX OKPY>KHOCTeH
123. PaccToHHHe Me>KAY TOllKaMH A H B paBHo 10. KaKoH HaH6oJlb-
WHH paJlHYc MO>KeT HMeTb OKPY>KHOCTb c UeHTpOM B TOllKe B, eCJ1H OHa
KacaerCH OKPY)KHOCTH c u.eHTpOM B TOlJKe A pallHyca 7?
124. PaccToflHHe Me>K!lY TOtlKaMH C H D f>aBI-Io 9. KaKoH HaHMeHb-
WHii pallHYc MO>Ke.r HMeTb OKPY>KHOCTb c ueHTpoM B TOl!Ke D, eCJlH OHa
KacaercSl OKPY)KHOCTH C lleHTpOM B T04Ke C pallHyca 5 '?
125. PaCCT05l11He Me>KllY TOlJKaMH E H F pasHo 6. KaKoH H3HOOJlb-
WHH paHYc MOJKeT HMeTb OKPY>KHOCTb C Llel-iTpOM B TOIIKe F, eCJ1H OHa
KacaeTCH OKPY>KHOCTH c ueHTpoM B TOIIKe E pallHyca 8?
126. PaccToHHHe Me)KJ1Y T04KaMH P H Q paBHo 4. KaKoH HaHMeHb-
WHH paJIJfYc MO>KeT HMeTb OKPY>KI-IOCTb C lleHTpoM B TOllKe Q, eCJII1 OHa
KacaeTCH OKPY>KHOCTH c ueHTpOM B TOllKe P paJ1Hyca II'!
127. -Use OKPY>KHOCTH pallHycoB 5 H 12 KaC310TCH llpyr Jlpyra B TOll-
Ke C BHyrpeHHHM o6pa30M (T.e. MeHbW3H OKPY)f{HOCTb paCn0J10IKel-la
BllYTPH 60JlbWCtf). 4epe3 TOllKY C npOBeJ.lel-l3 npHMaSl, nepeCCK81OllI.aH
60JIbWYlO OKPY)I(HOCTb B TOIIKe A, a MeHbWYIO OKPY>KHOCTb - B T04Ke
B. I-IaHn.HTe llJIHHY oTpe3Ka AB, eCJlH BG - 7.
128. nse OKPY)I{HOCTH paJtHycoB 13 H 17 KacalOTCSI Jlpyr !lpyra B T04Ke
C BHYTpeHHHM o6pa30M. 4epe3 TOIIKY C npoBe.aeHa IlpHM3S1, flepece-
KalO1ll8H 60JIbWYIO OKPY>KHOCTb B TOIIKe A, a MeHblUYIO OKPY>KHOCTb
B TOLJKe B. l-iaHJtHTe J1JJHHY oTpe3Ka BC, ec.nH AB 6.
129. llBe OKPY)I{HOCTH paAHYCOB r H R (R > r) KaC310TCH l1pyr .npyra
B TOlJKe C BHYTpeHHHM o6pa30M. 4epe3 TOqKY C npoBen.eHa npHMaH,
nepeCeKalOLl.l3H 60JlbWYIO OKPY}f{HOCTb B TOIIKe A, a MeHbWYIO OKPYIK-
HOCTb - B TOlIKe B. H3HJ1HTe J1JJHHY OTpe3Ka AC, eCJ1H AB a.

flJlHHa OKPY>KHOCTH H n.nOmaJlb Kpyra
101
130. llBe OKPY>KHOCTH paJlHycoB r M R (R > r) K3calOTCH l1pyr J1pyra
B TDtJKe C BHeWHHM o6pa30M (T.e. HM OllHa 113 HHX He J1e>KHT BHYTP"
ltpyrOH). npOBen.eHa npHMaH, KOTopaH KacaeTCH aD-HOM OKPY>KHOCTH B
TOllKe A, a J1pyroH B T04Ke B. HaHllHTe llnHHY OTpe3Ka AB.
131. llBe OKPY>KHOCTH paJ1l1ycoB T H R (R > r) KacalOTCSI llpyr npy-
ra B TOLlKe C BHeWHHM o6pa30M. npOBeJleHbl JIBe npHMble, Ka>K,llaH 113
KOTOpblX K3caercSi o6eHx OKpY>KHOcTeft. O,QHa npHMaSi KacaeTCH 60Jlb-
weH OKPY>KHOCTH B T04Ke A, MeHbwew - B TOlJKe B, a llpyraH npHMaH
KacaeTCH MeHbweH OKPY)KHOCTH B TOllKe C, a 60JIbweCt - B TOLJKe D.
nYCTb P T04Ka nepeCelJeHHH npHMblX AB H CD. Hai1D.MTe JIJ1HHY
orpe3K3 AP.
132. C l II1TaH, 4TO B YCJJOBHH npellblJlYllleH 3a!la l lH pal1HYCbl OKPY>K-
HocreH paBHbl 7 H 15, HaHJlHTe JJ.JIHHY orpe3Ka PO, r}le 0 LleHTp
u u
MeHbweH H3 OKPYJKHOCTeH.
133.* B Tpeyro.nbHHKe ABC yrOJI npH BepwHHe C npHMoH, AB 17,
BC 8. HaHJlHTe pallHYc OKPY/KHOCTH, Kac310LlleHcSI npHMblx AB, AC
u u u
H K3calOllleHCSI OKPY>KHOCTH, BnHcaHHOH B llaHIblH Tpeyr0J1bHI1K.
134.* B TpaneLlHIO ABCD BnHcaH3 OKPY)KHOCTb, KacalOlllaHCH 60KO-
BOH CTOpOHbl AB B TOLJKe At! TaKOH, liTO BJ\;I: AM = 1: 4. 113BecTHo,
liTO AB 10, AD 24. HaHllHTe paJlHYc OKPY>KHOCTH, KaCalOll.leHCH
npSlMbIx AD H CD H KacalOllleHcSI OKPY>KHOCTH, BnHC3HHOH B 118HHYIO
rpaneLlHIO.
7 .9. AnHHa OKPY>KHOCTH Ii nJlOl.U-a,nb Kpyra
135. Ha KJleT l l3TOH 6YMare c pa3MepOM KJ1eTOK
lCM x lCM H306pa>KeH Kpyr, llaCTb KOToporo 33WTPH-
XOB3Ha. Haii.D.HTe nJIOLll3)lb S 3awTpHxoBaHHoH LJaCTH
Kpyra B KB3.llpaTHbiX caHTHMeTpax. B OTBeTe 3anHWH-
S
Te 7r .

102 Z OKPYHOCTb
.
136. I-Ia KJ1eT l laT011 6YMare c pa3MepoM KJIeTOK
lCM x lCM H306pa}l{eH Kpyr, lJaCTb KOToporo 3awTpH-
xosaHa. HaHn.liTe nJlO1l1311b S 3awTpHxoB3HHOH LJ3CTH
Kpyra B KBallpaTHbIX c3HTHMeTpax. B OTBeTe 3anHWH-
S
Te 7r .
137. H3 K.ne.TtlaTOH 6YMare c pa3MepoM KJIerOK
lCM x lCM H306paJKeH Kpyr, lJaCTb KOToporo 33LUTpM-
XOB3Ha. HaHJ1HTe nJlOLuallb S 3awTpHxoB3HHOH llaCTH
Kpyra B I{BaJIpaTHbIX caHTHMeTpax. B OTBeTe 3anHlliH-
S
Te 7r .
138. 1a KJlelllaTOH 6YMare c pa3McpOM KJIeTOK
lCM x lCM H306pa)KeH Kpyr, lJaCTb KOToporo 3aulTpH-
XOB3H3. HaHJlHTe nJlOwallb S 3awTpHxoBaHHoH lJ3CTH
Kpyra B KB31lpaTHbiX Cal-lTHMeTpax. B OTBeTe 3anHWH-
S
Te 7r .
139. Ha Kf]eTtlaTOH 6YMare H306pa)l{eH yroJl LABC, CM. llaHHbIH 1-111-
>Ke PHCYI-IOK. I--IaHltHTe n.nOllla,nb S nepece l leHli51 yr.na LABC H Kpyra
C Ilel1TpOM B TOllKe B paJlHyca 18. B OTBere 3anHWHTe  .
B A
140. Ha K.neT4aTOH 6YMare H306pa>KeH yroJl LABC, CM. n.aHHbIH HH-
>Ke PI1CYHOK. HaHllHTe n.nOlllaHb S nepece4eHHH yrJla LABC H Kpyra
c uel-iTpOM B T04Ke B pallHyca 0,8. B OTBeTe 3anHWHTe ; .
B A
141. llaHblllBe KOHueHTpHlIecKHe OKPYJKHOCTH (T.e. OKPY>KHOCTH, ueH-
TPOM I{OTOPbIX HBJl5IeTC5I 011H3 H Ta )l(e TOLJKa). P3JlHYC 6o.nbweH OKPY)f{-

ll.nHHa OKPY)/(HOCTH H nJlOlllaJI.b Kpyra
103
HOCTH paBeH 20, a pa.QIiYc MeHbweH OKPY>KHOCTH paBeH 16. HaHJlHTe
nJ10111a,llb KOJlbua, OrpaHMlJeHHOrO 3THMH OKPY>KHOCTHMH. B OTBeTe 3a-
B
nHWHTe IT .
142.llaHbI llBe KOHLleHTpHLJeCKHe OKPY>KHOCTH (T.e. OKPY>KHOCTH, ueH-
TpOM KOTOpblX HBJ1HeTCH OIlHa H Ta JKe TOllKa). PanJ1Yc 60JlbWeH OKPY>K-
HOCTH paBeH 1, a pallHYc MeHbweH OKPY}f{HOCTH paBelt 0,5. HaHD.HTe
nJIOlllaJ1b KOJlbLl3, OrpaHHlJeHHOrO 3THMH OKPY>KHOCTHMH. B OTBeTe 3a-
B
nHWHTe 7r .
143. nJ10111a!lb KOJlbua, orpaHHtleHHoro JlBYMH KOHueHTpHlIecKHMH
OKPY>KHOCTS1MH, paSHa 967r, a paltHYc MeHbweH H3 OKPY>I(HOCTeH pa-
BeH 10. HaMllHTe pallHYc 6oJlbwei1 H3 OKpY>KHOCTeii.
144. nJlOUlaJ1.b KOJ1bua, orpaHH4eHHoro D.BYM5I KOHueHTpHlJeCKHMH
OKPY>KHOCTSlMH, paSHa 677r, a pallHYC 60JlbWeH 113 OKpY>KHOCTeH pa-
BeH 34. HaHJlI1Te P311HYC MeHbweH H3 OKPY>KHOCTeH.
145. nJlOUlaJJ.b KOJI, bLla orpaHH4eHHoro llBYMSI KOHLleHTpHLJeCKHMH
OKPY>KHOCTHMH, paSHa 3697r, a paJ1.HYCbl 3THX OKpY>KHOCTeH OTHOCSlT-
CH K3K 20 : 21. HaHJlI1Te paHHYc 6oJlbwei1 113 OKPYiKHOCTeH.
146. nJ10Ula.Qb KOJIbua, OrpaHHLJeHHOrO JlBYMSI KOHueHTpHlJeCKHMH
OKPYiKHOCTHMH, pasHa 3361r, a paJlHYCbI 3THX oKpymHocTeH OTHOCHT-
CH KaK 2 : 5. HaHJlHTe paJlMYc MeHbllIeH 113 OKpY>KHOCTeH.
147. nJ10maJtb KOJlbua, OrpaHHlJeHHOrO )!BYMH KOHueHTpHQeCKHMH
OKPY>KHOCTHMH, pasHa 80001r, a nJ10Ul3JlH 3THX OKPY)KHOCTeH OTHOCHT-
CH KaK 1 : 6. HaHllHTe paJlHYc MeHbweH H3 OKpY>KHOCreH.
148. nJlOllla,Qb KoJIbLla, OrpaHHlJeHHOrO JlBYMH KOHl.leHTpHtlecKHMH
OKPY>KHOCTSlMH, paBHa 36007r, a nJlOlU311H 3THX OKPY>KHOCTeH OTHOCHT-
Cst KaK 3 : 4. HaHJtHTe pa,llHYc 60JlbllleM H3 OKpY>KHOCTeH.
149. HaHJlHTe llJIHHY JlyrM ceKTopa Kpyra, paAHYC KOToporo paBeH 30,
eCJlH H3BeCTHO, liTO nJIOllla.nb 3Toro ceKTopa paSHa 180.

104
7. OKPY)/(HOCTb
.
150. HaHHHTe D.JIHHY llyrM ceKTopa Kpyra, paLlHYC KOToporo paBeH 0,5,
eCJlH H3BeCTHO, 4TO nJlOlllallb 3Toro ceKTopa paBH3 0,4.
151. HaHJl.HTe nJlOU.la.nb ceKTopa Kpyra, paJJJiYC KOToporo paBeH 15,
eCJlH H3BeCTHO, LITO JlJlHHa llyrH 3TOrO ceKTopa paSHa 20.
152. HaHJ!HTe nJlOlllaJlb ceKTopa Kpyra, pan.HYC KOToporo paBeH 16,
ec.nH H3BeCTHO, liTO llJlHHa !lyrH 3Toro ceKTopa paBHa 21.
7.1 o. 3aaLJI1 H8 OKa3aTeJlbCTBO
153. B MI-IoroyroJlbHHK BIIHcaHa OK)Y)l{HOCTb pallHyca T. llOK3)KHTe,
4TO nepHMeTp 3Toro MHoroyrOJIbHHKa 60J1bWe, lleM 21fT.
154. BOKpyr MHoroyrOJIbHHKa onHcaHa OKPY)KHOCTb pallliyca R. llo-
Ka>KHTe, LITO nepHMeTp 3Toro MHoroyroJlbHl1Ka MeHbwe, lleM 27r R.
155. OKpY>KHOCTb K3caeTCS1 CTOpOHbl AB TpeyrOJlbHHKa ABC B T04-
Ke P 11 KacaeTCH np OD.OJl)KeHHH CTOpOH AC H BC 3a T04KH A H B.
156. Bnl1C3HHaSl OKPY>KHOCTb TpeyroJlbHH K a ABC KaCaeTC5J e ra CTO-
.
A
157. TOlJKa 0 uel1Tp BnHcaHHOH OKPY>KHOCTH
Tpeyro.nbHI1Ka ABC.  OKa>KH Te , liTO
LAOB
158. TOIIKa 0 LleHTp OKPY>KHOCTH, K3calOllleH-
CH CTOpOHbl AB H npOJlOJI>KeHHH CTOpOH AC H BC
TpeyronbHHKa A B C, CM. P HCYHOK. lloKa>KMTe, qTO
LAOB

3aJJ.alJII Ha Jl-OK83aTeJlbCTBO
105
159. BnHcaHHaSl OKPY>KHOCTb TpeyroJlbHHKa
ABC K3CaeTC5I era CTOpOH B TOlIK3X AI, B 1 ,
C 1 , CM. P"ICYHOK. ,L:loKa>K H Te, 4TO
A
B I
B
c
t 60. llaH TpeyrOJIbHHK ABC H OKPY>K-
HOCTb, KOTopaS1 K3caerCH CTOpOHbl AC
11 npOJlOJI)I{eHHH CTOpOH AB H BC B
TOllKax B 1 , C 1 , Al COOTBe TC TBeHHO.
161. ,UoKa>KHTe, LITO eCJ1H B 4eTbIpexyroJlbHHK ABC D BnHcaH3
OKPYiKf-lOCTb C LleHTpOM B TOllKe 0, TO LAO B + LCO D 180 0 .
162. ,UoKa>KHTe, 4TO eCJIH B wecTHyrOJ1bHHK ABCDEF BnHcaHa
OKPY>KHOCTb c uel-fTpOM B TOllKe 0, TO LAOB+LCOD+LEOF 180°.
C
163. B wecTHyrOJIbHHK ABCDEF BnHC3Ha
OKPY>KHOCTb, CM. PHCYI.JOK. nOK3IKHTe, 4TO
AB+ CD+EF BC+DE+AF.
A
D
164. WecTHyronbHHK ABCDEF BnHC8H B
OKPY)I{HOCTb, CM. PHCYHOK. lloKa>KHTe, liTO
LA + LC + LE 360°.
F
C
165. J(Be OKPY>KHOCTH KaCaIOTC5I Jlpyr llpyra BHeWHHM o6pa30M B TOlJ-
Ke C. K 3THM JtBYM OKPY>KHOCTHM npOBelleHa o6lUaSi KaC3TeJlbHaSl, K3-
C3IOlU3S1CH OJlHOH 113 HHX B TOI.fKe A, a n.pyroM B TOllKe B, CM. PHCY-
HOK. J(OK3>KHTe, lJTO LAC B 90 0 .

106
7. OKPYJKHOCTb
,.
166. nBe OKPY>KHOCTH KacalOTCH n.pyr llpyra BHeWHHM o6pa30M B T04-
Ke C. K 3THM LlBYM OKPY)KHOCTHM npOBeJleHa o6111aH KaCaTeJ1bHaH, Ka-
calOlllaHCH OJtHOH H3 HHX B TOtlKe A, a BTOpOH B l'O'-lKe B. npflM3S1
BC nepeceKaer nepBYIO OKPY>KHOCTb B TOllKe D, CM. PHcYHoK.lloKa-
>KliTe. liTO OTpe30K AD SlBJ1HerCH !lHaMeTpoM nepsoH OKPY)KHOCTH.
167. ,.Use OKPY>KI-IOCTH KacalOTCH BHyrpeHHHM o6pa-
30M B TOIJKe A. Ha 60JIbwei1 OKPY>KHOCTH B3HJ1H TOtlKH
B H C 11 npOBeJ1H XOpJIbl AB MAC, KOTopble nepe-
ceKalOT MeHbwylO OKPY>KHOCTb B T04Kax D H E COOT-
BeTcTseHHo, CM. PHCYHOK. nOK3JKHTe 410 npSlMble BC
H DE napaJIJIeJlbHbI.
A
168. nBe OKPY)KHOCTH K3caIOTCS1 !lpyr .npyra BHeWHHM o6pa30M B TOLJ-
Ke A. 4epe3 TOtlKY A npOBeJlel-lbI )tHe npHMble, nepeceKalOlllHe BTOpH4-
HO O.QHY 113 OKpY>KHOCTeH B TOllKax B H C, a llPYrylO B T04Kax D H
E, CM. PHCYHOK.lloKa)l(HTe, qTO npHMbIe BC H DE napaJlJIeJlbHbI.
B
E
169. ,Use OKPY>KHOCTH nepeceKalOTcSI B T04K3X P H Q. 4epe3 TOlJKH

3a,aalfH Ha J1.0Ka3aTe.J1bCTBO
107
P H Q npOBelleHbl npHMbIe, nepeCeK310UlHe OD.HY H3 OKPYIKHOCTeH B
T04K3X A H C, a JlPYrylO B TOIJKaX B 11 D, CM. PHCYHOK. llOKa)l{HTe,
liTO npHMbIe AC H BD napa.nJleJJbHbl.
A
170. llBe OKPY>KHOCTH nepeceKalOTcH B TOlJKaX P H Q. 4epe3 TOllKY
A nepBoH OKPY>KHOCTH, OTJIHlJHYIO OT TOlleK P H Q, npOSellel-lbl npH-
Mble, nepeceKalOLUHe BTOpYIO OKPY>KHOCTb B TOllKax B H C, CM. PH-
CYHOK. llOKa)l{HTe, liTO KacaTeJJbH3S1 B TOlJKe A I< nepBol1 OKPY>KI-IOCTH
napaJlJIeJIbHa npHMoH BC.
B
c
171. TOQK3 P - ueHTp BnHcaHHOH OKPY)I{-
HOCTH TpeyrOJ1bHHKa ABC. JlY4 AP nepe-
ceKaeT onHcaHHYIO OKPY>KHOCTb TpeyroJlbHH-
Ka ABC B TOlJKe A l , CM. PHCYHOK. ,iloK3>KHTe,
4TO PAl BAl.
A
Al
172. DHcceKTpHca BHeWHero yrJI3 npH-
BepWHHe C TpeyronbHHKa ABC nepece-
KaeT onHcaHHYIO OKPY>KHOCTb TpeyrOJIbHH-
K3 B TOlJKe D, CM. PHCYHOK. ll.oKa}KMTe, liTO
AD BD.
A

108

,\, , "'1 'I ",". It -'I.
'.
173. l{oKa>KHTe, lfTO eCJlli ')'()III<lt (' II I' ,11,',1,.11 lilt ".'IIIY 'IC.I)lHIY ()I
nps1MoH AB H LACB = LrtJ)J.l 'J(J Me),I,llft IlIlItIU'" III 1)1\1''/I\II()l'rl.,
COLlep)f{awylO Bce lleTblpc T()lIKII ./1  / I, ( '. I}
174. llOKa>KHTe, liTO eCJlli l"YMM:1 lll'"\ 1IIIIIIIIIICIIUt.IUtil\III.IX YI'.'I()III\I)I-
nYKJlOrO 4eTblpexyroJlbHHK:l r),11411.1 11)". I.. I")I\I'}I' 11('1 () Mc)}I<II() ()III'I-
caTb OKPY}KHOCTb.
175. nOKa)KHTe, lITO CCJIli Jl..fISIIU.IIIYI\.f'(JI CJ 'Il'II.II)lxyr()JII)1111Ka ABCD
CnpaBeDJIHBO paBeHcTBo An -f- (fJ) lIt', /1/), '1'0 Ii TOTlleTblpex-
yrOJIbHHK MO>KHO BnHC3Tb ()KI)YikIIC)l' II..
176. nYCTb H - TOtlKB Ilr')t'l'l'II('IIII1 "I.ll'()'1 'r()eyrOJ1bHHKa ABC, a
K, L, 1\;[ - TOlIKH, CHMMC1-11Iilllll.ll' I (JIII\(' II (),I'II()CHTeJIbHO CTOpOH AB,
BC HAC. nOK3>K11Te. lIT(> T()'IKII 1\', I., J\ I .IICiKltT Ha onHcaHHOH OKPY)I{-
HOCTH Tpeyr'onbHHKa ABG-Y.
177. nYCTb H T04Ka Ill'))l'l'l"lt'IIII)1 I\I)IC()T AA 1 , BB l , CC I Tpey-
rOJ1bHHKa ABC. nOKamHTc, 'rr() -I C )'11\.1 If SIIJJ1HeTCS1 ueHTpoM BnHC3H-
HOM OKPY>KHOCTI1 TpeyrOJII)lili 1(,1 ;, I J II ( fl.
178.* B rpeyrOJIbHHKe ABC IIII ())llill ILi yr.l1()B He HBJ1HerCH npHMblM.
OTpe3KH AA 1 , BB], CC l - lil)lt'(rl'l.1 '1'1)('yr()JII)1111Ka ABC, H - TOqKa
nepeCelJeHHH 3THX BbICOT. llOKC:lil{lrrl', IIT(, lrl)eJlHHbl CTOpOH rpeyrOJIb-
HHKa ABC H CepeJ1HHbl OTp;il«)I' ./\//, '_Ill, (-;H Jle>KaT Ha onHcaH-
HOH OKPY)KHOCTH TpeyrOJIblfIK1 ill J J I ( f I ( -J 1';1 ()I([)Y)KHOCTb )-{a3bIBaeTCSI
OKPY)f{HOCTblO 3iinepa MJlH OK[)Yil{II()l"II.f() )lCIrrlt TOlleK).
179.* TOllKa P UeHTp BnHCc:lllli()fl ()I\I)Yil\II()lrt TI)eyrOJ1bHHKa ABC.
npHM8H AP nepeceKaer onHccll-IIIYU) ()KI)Yil\II()C'J'I) rpeyrOJ1bHHKa ABC
B TOQKe Ale nOK3)f{HTe, 4TO Ap. [)j-1 1   Ii · /', rllc R - paJlHYC onH-
caHHOH OKPYJKHOCTH TpeyrOJIbilliKc1 JllJ(., ;1 ,. = 1):)JlIYC era BnHcaHHoH
OKPY)f{HOCTH.
1"80.* nYCTb R pa,nHYC onHCaHI-iofl (}KI)YiKll()CT1 lleKOToporo rpe-
yrOJlbHHKa, r pa.aHYC ero BnHCal-lHOi-i ()KI)Y>I<II()C'('H, a d paCCTOH-
HHe Me}l{JlY ueHTp3MH 3THX OKpY>KHOCreH. Lt()Kil)f{Ji're, liTO cnpaBeMHBO

KOHTpOJlbHan pa60ra I 09
CJlen-YlOlllee paBeHcTBo: d 2 R 2 - 2R. r (3TO COOTHoweHHe Ha3blBa-
eTCH cpOPMYJlOH 3HJ1epa).
KOHTpOJlbHaS1 pa60Ta.
PeKoMeHllyeMoe BpeMst J]JJ5I BblnOJ1HeHHH pa60Tbi 40 MHHYT.
PeKoMelfLlyeMa51 WKaJ1a OlleliHBaHHH pe3YJlbTaToB:
'-" 0-1 2 3 4
'-IIIICJ10 aepHblX peweHIIIM
WKOnbHa$l 0u.eHKa 2 3 4 5
I
BapHaHT 1
1. Xopl!a AB JleJIl1T OKPY)KHOCTb Ha JIBe llyrH, rpaIlYCHble BeJJHLJHHbl
KOTOpblX OTHOCHTCH KaK 4 : 11. no.n KaKHM yrJ10M BH)lHa 3Ta XopJla "3
T04KH C, npHHallJle)J{ameH MeHbweH ltyre OKpY>KHOCTH? OTBeT ltaHTe B
rpaHycax.
2. B OKPY>KHOCTH, llHaMeTp KOTOpOH paBeH 34, npoBelleHa xoplta, T3-
KaH, liTO paCCTOHHHe OT ueHTpa OKPYiI(HOCTH 110 3TOH XOpllbl pasHo 8.
laHllHTe ,ll.J1HHY 3TOH XOpllbl.
3. CropoHa pOM6a paSHa 4J3, a ocrpbIH yrOJ1 pOM6a paBeH 60°. HaH-
JlHTe paD.HYC BnHC3HHOH OKPYIKHOCTH pOM6a.
4. nepHMerp tieTbipexyrOJIbHHK3 paBeH 62, Q)lH3 H3 era CTOpOH pasHa
20, a -npyraSi - 15. HaHllHTe 60JIbWYlO 113 OCTaBlllHXCH CTOpOH 3Toro
..
lleTblpexyrOJIbltHKa, eCJ1H H3BeCTHo, liTO B Hero MO>KHO BnHcaTb OKPY)K-
HOCTb.
5. nJlOllla.nb KOJJbua, KOTopoe orpaHH4eHO llBYMSI KOHueHTpHQeCKMMH
OKPY)KHOCTHMM, paBHB 31001r, a pa-nHYCbl OKPY>KHOCTeH OTHOCSITCH KaK
15 : 16. HaHllHTe pan.I1Yc 60JlbWeH 113 OKpY>KHOCTeH.

110
9 7. ()K/JY)/(H()CTb
..'
BapHaHT 2
1. XopJla AB ,LleJ1I1T oKpymHoCTI) 11,1 JlBC Jlyr1, rpallYCHble BeJ1HlJHHbl
KOTOpblX OTHOCSITCH KaK 7 : 23. n{)Jl K<IKltM yrJ1()M BHllHa 3Ta xop.na H3
TOllKH C, npHH3llJ1e>K3LlleCt 60JlbllJefi llyre OKPY>KHOCTH? ()TseT llaHTe B
rpan.ycax.
2. B OKPY)KHOCTH C Jll1aMeTpOM, paBI1bIM 26, IlpOBel1eHa xoplla, llJlH-
H3 KOTOpOH paBHa 10. HaHllHTe paCCT05lHHe OT lleHTpa OKPY>KHOCTH .no
u
3TOH XOPJlbl.
3. OCTpblH yroJi pOM6a paoel1 30°  a pallHYc BnHcaHHOH OKPYIKHOCTH
3Toro pOM6a paBeH 16. HaHJll1Te CTOPOHY pOM6a.
4. nepl1McTp 1leTblpexyr0J1bl1HKa paBeH 66 OltHa 113 era CTOpOH paSHa
16, a Llpyrafl - 20. HaHllHTe MelbUJYIO 113 OCTaBWHXC5I CTOpOH 3T()I'O
..
lJeTblpexyrOJlbHHK3, eCJ1H 113ReCTHO, liTO B Hero MO}KHO BnHcaTb OKPY)f{-
HOCTb.
5. nJ10Illa.nb KOJIbua, KOTopoe orpaHHtleHO llSYMst KOI-lueHTpH4eCKHMH
OKPY>KHOCTHMH, paSHa 5607r, a pa}]JiYCbl OKPY/KI-fOCTeH OTHOCITCs-t KaK
17 : 18. HaMJlHTe paJlHYc MeHbllleti H3 OKpY)KHOcTeti.
BapHaHT 3
I. T01.JK3 0 ueHTp onHcaHHOH OKPY>KHOCTH
TpeyroJlbHHKa ABC, yrJlbl AGB H AOC paBHbl
114 0 H 116 0 COOTBerCTBeHI10. I13HllHTe rpaLlYc-
HYIO Mepy yrJla A TpeyrOJ1bHHKa ABC.
2. B OKPY>KHOCTH npOBe!leHbl XOpJ!bl AB H CD. HaHJ!I1Te paccToHHHe
OT ueHTpa OKPY*HOCTH llO XOPllbI CD, eCJ1H AB 10, CD 24, a
paccToHHMe OT lleHTpa OKPY)I{HOCTH .no XOPJlbI AB paBHo 12.

KOHTpOJTbH8n pa60Ta
I I 1
3. 50KOBbIe CTOpOHbl paBHo6en.peHHoro TpeyroJlbHHK3 paBHbl 5, a oc-
HosaHHe paBHo 6. HaHHHTe paAMYc BnHc3HHoH OKPY>KHOCTM TpeyrOJ1b-
HHKa.
4. H3BecTHo, lITO B lleTblpexyrOJIbHHK BOHcaHa OKPY>KHOCTb, a JlJ1HHbl
Tpex 113 CTOpOH 3Toro 4eTbIpexyroJlbHHKa paBHbl 2, 4, 8. KaKoe l-IaH-
60J1bWee 3Ha l ieliHe MOlKer HMeTb 1lJ1liHa LJeTBeproM CTOpOHbI 3Toro lJe-
TbIpexyroJlbHHKa?
5. nJlOll.lan.b KOJ1bua, KOTopoe OrpaHHlJeHO llBYMH KOHueHTpHlJeCKHMH
OKPY*HOCTHMH, paSHa 20161r, a paAI1Yc MeHbweM 113 3THX OKPY*HO-
CTeH paBeH 3. HaH,QHTe pallHYc 60JlbWeH 113 OKpYiKHOCTeH.
BapHaHT 4
1. T04K3 0 - ueHTp onHcaHHoH OKPY>KHOCTH rpe-
yrOJIbHHK3 ABC, yr JIb) AOB H AOC paBHbl
99° H 101 0 COOTBeTCTBeHHO. HaHHTe rpayc-
HYIO Mepy yrJIa A TpeyroJlbHHK3 ABC.
A
2. B OKPY)I(HOCTH npOBelleHbl XOpllbl AB H CD. HaHJlHTe paccToHHHe
OT ueHTpa OKPY)I{HOCTH AO XOP)!bl CD, eCJ1H AB 80, CD 18, a
paccToHHHe OT LleHTpa OKPY)I{HOCTM JiO XOpJlbl AB pasHo 9.
3. 50KOBble CTOpOHbl paBHo6ellpeHHoro TpeyroJlbHHKa paBHbl 17, a
OCHOBaHHe pasHo 16. Hatf.QHTe paHHYc BnHcaHHoM OKPY)KHOCTM rpe-
yro.JlbHHK3.
4. 113BecTHo, 4TO B lJeTblpexyroJlbHHK BnHcaHa OKPY>KHOCTb, a !lJIH-
Hbl Tpex H3 CTOpOH 3Toro 4eTbIpexyrOJIbHHKa paBHbl 13, 15, 17. KaKoe
.. U
HaHMeHbwee 3H3 t leHHe MO}KeT HMeTb AJlHH3 lJeTBepToH CTOpOHbl 3Toro
qeTblpexyrOJIbHHKa?
5. nJJOll.lallb KOJlbua, KOTopoe orpaHH4eHO JlBYMSI KOHLleHTpH1JeCKHMH
OKPY)KHOCTHMI1, pasHa 20157r, a paAHYc 60JIbllleH 113 3THX OKPY>KHO-
CTeH paBel-l 48. H3HDJiTe P311HYC MeHbweH 113 OKPY>KHOCTeH.

112
.
 7. OKPY»<HOCTb
BapHaHT 5
I. LIeTblpexyro.flbHHK ABCD BnHcaH B OKPY>KHOCTb. Vl3BecTHo, 4TO
LCD B 44° H LABD 45°. HaHllHTe rpallycHYIO Mepy yrJIa LBPC,
rJ1e P - TOlJK3 nepeCelJeHH5I LtMarOHaJIeH lJeTblpexyrOJIbHHKa ABC D.
2. OTpe30K AB S1BJ1HeTCH XOp,aOH OKPY)KHOCTH c ueHTpOM B T04Ke O.
TOIJKa P, JIe>Kalllafi Ha OTpe3Ke AB, TaKOBa, 4TO AP 8, 0 P 10,
BP 12. HaHJlHTe pa-fiHYc OKPY>KHOCTH.
3. fHnoTeHY3a npHMoyrOJ1bHOrO TpeyrOJ1bHHKa paSHa 26, a paJlHYC BnH-
caHHOM OKPY)KHOCTH 310ro Tpeyr0J1bHHK3 paBeH 4. HaHJlHTe nepHMeTp
3Toro TpeyroJlbHHKa.
4. TpH CTOpOHbl onHcaHHoro OKOJIO OKPY>KHOCTH lJeTblpexyrOJlbHHKa,
B3ffTble B nOCJ1e,llOB3TeJIbHOM nopHllKe, OTI-IOCHTCH KaK 15 : 16 : 20.
HaHJlHTe 60JlbWYIO 113 CTOpOH 3Toro 1.JeTblpexyrOJlbHHKa, eCJlH era ne-
pHMeTp paBeH 21.
5. HaHJlHTe ){JlHHY JlyrH ceKTopa Kpyra, paLlHYC KOToporo paBel-1 5, eCJ1H
H3BeCTHO, I.JTO nJlO1ll3Jlb 3Toro ceKTopa paBHa 55.
BapHaHT 6
1. 4eTblpexyrOJIbHHK ABCD BnHcaH B OKPY)KHOCTb. H3BeCTIIO, LITO
LCDB 25° H LACD 50°. HaHl1HTe rpaJ1ycHYIO MepyyrJla LBQC,
fJle Q TOlIK3 nepeCelJeHHH npOJlOJl>KeHHH CTOpOH AB 11 CD.
2. OTpe30K AB S1BJl5IeTCH XOPJlOH OKPY)KHOCTH C ueHTpoM B TOllKe o.
TOlJK3 P, Jle>K31113H H3 oTpe3Ke AB, T3KOBa, liTO AP 15, OP 13,
BP 21. HaHJ{HTe paJtHYc OKPY>KHOCTH.
3. fMnoTeHY3a npSlMoyrOJIbHOrO TpeyrOJIbHMKa paSHa 35, a paltHYC snH-
C3HHoii OKPY>KHOCTH 3Toro TpeyrOJIbHMK3 paBeH 7. HaHlI.HTe nepHMeTp
3Toro TpeyroJlbHHKa.

KOHTpO.l1bHafi pc160Ta
113
4. TpH CTOpOHbl onHC3HHoro OKOJIO OKPY>KHOCTH qeTblpexyroJlbHHKa
B3HTble B nOCJleJtOBaTeJlbHOM nopHllKe, OTHOCHTCSI K3K 11 : 12 : 14.
HaHllHTe MeHbwylO H3 CTOpOH 3Toro lJeTblpexyronbHHKa, eCJJH ero ne-
pHMeTp paBeH 56.
5. HaHllHTe JL1IHIY ll)'rH ceKTopa Kpyra, pa.QHYC KOToporo paBeH 6, eCJ1H
H3seCTHO, 4TO nJlOlLlaJlb 3Toro ceKTopa pasHa 78.
BapHaHT 7
1. 4eTbipexyroJlbHHK ABCD BnHcaH B OKPY)I{HOCTb. YroJI ABC pa-
SCH 135°, yroJl ADB paBeH 25°. HatiJtHTe rpal1ycHYIO Mepy yrJl3 BAC.
2. ,QaH3 OKPY>KHOCTb c ueHTpoM B TOlJKe 0 pan.Hyca 15 H TD1JKa A, pac-
CTOHHHe OT KOTOpOH JlO TOLJKH 0 pasHo 25. 113 TOLJKH A npoBelleHblllBe
np5lMble, K3C31OILlHeCH Jl3HHOH OKPY)I{HOCTH B TOlJKaX M H N. HaHJlHTe
MHHY oTpe3Ka MN.
3. DOKOBble CTOpOHbl paBHo6en.peHHoro TpeyrOJIbHHKa paBHbI 39, a
OCHOB3HHe pasHo 72. HaHltHTe pa1{MYc onHc3HHoH OKPY>KHOCTH 3Toro
TpeyroJlbHHK3.
4. llJIH CTOpOH lleTbIpexyronbHHKa ABCD, onHcaHHoro OKOJIO OKPY}I{-
HOCTH, BbInOJIHeHbl COOTHoweHHH: AB : BC 16 : 15, AD: CD
21 : 20. nepHMeTp lJeTblpexyrOJIbHI1K3 ABCD paseH 2016. HaHllH-
..
Te MeHbwylO H3 CTOpOH 3Toro QeTbIpexyrOJIbHHK3.
5. HaHllHTe nJlOlllallb ceKTopa Kpyra, pal1HYC KOToporo paBeH 10, eCJIH
113BeCTHO, LITO HJIHHa JlyrH 3Toro ceKTopa paBH3 11.
BapHaHT 8
1. 4eTbipexyroJlbHHK ABCD BnHcaH B OKPY)I{HOCTb. Yron ABC pa-
BeH 1450, yroJl DAC paBeH 600. HaHJtHTe rpaJIYCHYIO Mepy yrJla ACD.

114
7. ()KpY)I(HOCTb
2. LlaH3 OKPY)I{HOCTb C LleHTpoM B TOllKe 0 paJlHyca 14 H T04Ka A, pac-
CTOHHHe OT KOTOpoCl 110 T04KH 0 pasHo 50. 113 TOllKH A npoBelleHbl JIBe
npSlMbIe, K3calOlllHeCH JlaHHOH OKPY>KHOCTH B T04Kax ]}I H N. HaHJlHTe
DJIHHY OTpe3Ka MN.
3. DOKoBble CTOpOHbl paBH06eJlpeHHOro rpeyroJlbHHKa paBHbl 90, a pa-
llHYC onHc3HHoti OKPY>KHOCTH 3Toro TpeyrOJIbHHKa paBeH 75. HattllHTe
J1HHY OCHOBaHHH 3Toro TpeyroJlbHHKa.
4.llJlH CTOpOH lJeTbipexyrOJIbHHK3 ABCD, onHcaHHoro OKOJlO OKPY>K-
HOCTH, BbInOJlHeHbl COOTHoweHHH: AB : BC 15 : 17, AD : CD
11 : 13. nepHMeTp lJeTbIpexyrOJIbHHKa ABCD paBeH 336. HaHllHTe
60JIbWYIO 113 CTOpOH 3Toro lJeTblpexyrOJIbHHKa.
5. HaHJlMTe nJlOlllan.b ceKTopa Kpyra, paJlHYc KOToporo paBeH 15, eCJ1M
H3BeCTHO, liTO llJlHHa llyrH 3Toro ceKTopa paSHa 16.
BapHaHT 9*
1. OKpY>KHOCTb c ueHTpoM Ha cTopoHe AC TpeyroJlbHHKa ABC npo-
XOJlHT lIepe3 BepwHHY C H KacaeTCH npHMoH AB B TOIlKe B. HaHJlHTe
JtHaMeTp OKPY}I{HOCTH, eCRI1 AB 15, AC 20.
2. K OKPY)KHOCTH, BnHC8HHOH B TpeyroJlbHHK
ABC, npOBeJleH3 K3CaTeJ1bHaH, napaJlJ1eJ1bHaH
cTopoHe AB H nepeceKalOlUaH CTOpOHbl AC, BC
B TOlJKaX P H Q COOTBeTCTBeHHO. 113secTHo, liTO
AB 50, a nepHMeTp rpeyroJlbHHKa C PQ pa-
BeH 100. HaHDJiTe JlJlHHY oTpe3Ka PQ. A
3. nepHMeTp npHMoyrOJIbHOH TpaneUHH, onHcaHI10H OKOJIO OKPY>KHO-
CTH, paBeH 8, a MeHbwee H3 OCHOBaHHH pasHo 1. HaHllHTe pa.QHYc ,llaH-
u
HOH OKPYIKHOCTH.

/(OJITJ>OJlbH8n pa60r8
115
4. 4cTblpexyroJlbllHK ABCD BnHcaH B OKPY>KHOCTb. Vl3BeCTHO, liTO
AB 12, CD 16, AC 17, BD 18. HaHAHTe llJIHHY oTpe3K3
AP, r)lc P T01.JKa nepeCeqeHHst )lMarOH3J1eH AC H ED.
5. B Tpeyr-OJ1bHHKe ABC 6HcceKTpHca yrJIa A nepeceKaeT BbICOTY BH
B l-()(IKC P, npH 3TOM BP : PH 5: 4. Hal11lHTe paJlHYc onHcaHHOH
OKPY)I{lf O CTI1 TpeyrOJ1bl-lHKa ABC, eCJIH BC 21.
BapHaHTI0*
1. OKPY)f{HOCTb C [el-fTpOM H3 cTopoHe AC rpeyroJlbHliKa ABC npo-
XOllHT llCpC3 BepwHHY C H K3CaeTC5I npSlMoH AB B TOIIKe B. Hai:t.QHTe
LlH3Merp OKPY)KI-IOCTH, eCJlJ.1 AB 14, AC 16.
2. 1< ()KPY>KHOCTH, BnHcaHHoH B TpeyrOJ1bHHK C
ABC, npOBeLlel-l3 KaC8TeJlbHaH, napaJIJ1eJ1bH8H
cTopoHe AB J..I flepeceKalOlllaSl CTOpOHbl AC, BC
B TOllKax P H Q COOTBeTCTBeHHO. Vl3BeCTHO, liTO
AB 45, a nepHMeTp TpeyrOJIbliHKa ABC pa-
BeH 150. HaHJlI1Te llJlHHY oTpe3Ka PQ. A
3. nepHMeTp npSIMoyrOJIbHOH Tpanel.ll1H, onHC3HI-lOH OKOJlO OKPY>KHO-
CTH, paBeH 20 a 60JIbWee H3 oCHoBaHHH paBI-IO 7. HaH,llHTe pa!lHYc
OKPY}KltOCTH.
4. 4eTblpexyr0J1bHl1K ABCD BnHC3H B OKPY)I{HOCTb. 113BecTHo, 4TO
AB 18, CD 15, AC 26, BD 24. HaHl1HTe llJIHHY oTpe3Ka
AQ, f}te Q TOlIK3 nepeCelJeHHS1 npOJlOJI)KeHHH CTOpOH AB H CD.
5. B TpeyrOJIbHHKe ABC 6HcceKTpHca yrJ1a A nepeceKaeT Bblcory BH
B TOllKe P. HaHlll1Te OTHorneHHe BP : PH, eCJIH BC 60, a paJlHYc
onHc3HHoH OKPY>KHOCTH TpeyroJlbHHKa ABC paBeH 34.

116
PeWeHJfH K  1
.
PemeHHH K  1 (OCHoBHble nOHHTHH reoMeTpHH)
HOMepa 3aaa tl, K KomopblM npuBeaeHbl peUleHUJl: 5, 1 0, 23,
27,29,36,40,47,50,59,61,65,69, 75, 78,80,83,84,85,86,87,
88,89,91,92,93,94,95,97,98.
5. De3 orpaHHlleHHH OOLUHOCTI1 MO>K110 ClIHTaTb, liTO TOllKa B pacno-
J10>KeJ1a npaBee TOllKH A. TOllKCJ C, J1e>KBU-laS1 Ha npHMOi1 AB, MO}l<eT
6blTb paCnOJlOiKCI13 OTHOCHTeJlbHO TOllKH B llBYMH cnOC06aMH: rlpaBee
TOLIKH B iJ1H J1eBee TOlIKt B. Ha llaHHblX HHiKC PHCYHK3X I H 2 11306-
pa)l{eHbl oua 3TH CJ1Y l I351.
pac.]
A B
. .
34 45
c
.
pHc.2 C A 34
.
4S
B
-
B nepBoM CJ1Y4ae AC
cJlYLJae AC BC - AB
OmBem: 11 HJ1H 79.
AB + BC 34 + 45
45 - 34 11.
79, a BO BTOpOM
1 O. PaccMoTpHM llBa B03MO>KHbIX cJlYLJast pacnOJIO>KeHHH T04eK K Ii
L OTHOCHTeJlbHO KOI-lUOB oTpe3K3 !\t[ N:
I) TOlJKa [( Jle)KHT 6J1HII<e K TOqKe AI, lleM TOqKa L;
2) T04Ka I( .ne>KHT JlaJlbWe OT TOtlKH f, LJeM T04Ka L.
Ha J{aHHbIX HH}I{e PHCYI--IKax I 11 2 l.t306pa>Keltbl o6a 3TH CJlYllaH.
18
K L
. .
15
N
.
pHc.2 M L 15 K
'- T 
pRC. 1
21
M
.
B CJlY4ae 1) ItMeeM: AI N l K + K L + LN
B cnY4ae 2) HMeeM: 1\.[ L J\I! [( - K L 3, [( N
kIN IL + LI( + [(N 24.
OmBem: 24 HJlli 54
18 21
18+ 15+21
LN - ]( L
54.
6,
23. Ee3 orpaliHlJeHHH 06U-lHOCTH MO>KHO CLJHTaTb, tlTO T04Ka B pacno-
J10>Kela npaBee TOllKH A. TaK KaK AC : BC 5 : 3 > 1, TO B03MO>KHbl
TOJlbKO CJ1eJlYlOlllHe llB3 CJlYLJ351 pacnOJIOmeHHSI TOqKH C OTHOCHTeJIbHO
OTpe3K8 AB:

Ot'IIOIJIIIJ/() 11()1l,rl'lI 'II '('I JA1(\11JIIII
117
1 ) T()IIKc.1 ( JI('}KI,ll' 1\II}'TpI1 OTpe3Ka AB;
2) l'Otll<1 (1 ,/ll'il<ltT 11pcJIJCe TOllKH B, CM. PHCYHKH I If 2
c<)< )1'( 'e.rl'1' I JCI' II C).
5x
C B
e .
3x
pRc.2 A
...............
pHC. 1
B 3x C
. ---
A
e
Sx
n(Jl'l'I-lllI1IJII)II() 1()iMOIKI-fbIH CJlYllaH paCnOJIOiKeHHS1 TOLIKH C J1eBee
TOtll<li ./1 11:ll 1l ;IM()M JlCJIC 11C peaJ1H3yeTcH, T.K. B 3TOM cJ1Y1.Jae AC < BC,
CM. 1)lfC. :3, 'rr() "I)().fIIIJOpelIHT YCJ10BHIO AC: BC > 1.
puc.3 CAB
e ..
rlYl"rlj 13(0' - 3r, Torlla AC 5x. TOr!l3 B cJlY4ae 1) HMeeM:
All := l1(- + f3(- 8x 16 (no YCJIOBHJO AB 16), OTKYJt3 x 2,
BG' -: :iJ' == (). AllaJ10rH l lHo B CJ1YlJae 2) HMeeM: AB AC - BC
2J == 1 G, l. - 8, oTKyn.a BC 3x 24.
(),l1llenl: (j IJJI1 24.
27. rr;) I{ KCl K AD 25, CD 11, a pacnOJIO>KeHHe TOLJeK A, B, C, D
'1'(IK()C, KclK rl()1<333lI0 Ha pHCYHKe, TO AC + BD AB - CD 14,
18.
A M C
"'Le ,.
11
2S
D N B
. " e " -
OmBem: 18
29. 3aMeTHM, LITO nOCKOJlbKY pacnOJIO)l(eHHe TOlJeK A, B, C, D TaKoe,
KaK nOKa3BH O Ha pHCYHKe, T O AC + BD - AB + CD 80. Jl a Jlee
A DeB
e. . .
CJ1eJlOBaTeJ1bHO, M N AB - AM - BN 57 40 17.
OmBem: 17
36. B03MO)KHbl llBa cJlyttaH pacnOnO>KeHHH JIyqeH OA, OB " OC OT-
HOCHTeJlbHO J!pyr ,npyra:

118
PeweHHH K  1
I ) J1Y4 OA Jle)KHT BHe yrJl3 BOG, CM. PHCYHOK 1;
2) Jlyq OA JIe}KHT BHYTPH yrJIa BOG, CM. PHCYHOK 2.
pRC. 1
B
pHC. 2
B
A
A
o
Y1IHTbIBaH YCJIOBHe LADC < 180 0 , B nepBOM CJIyqae HMeeM:
LADG 360 0 - (LAOB + LBOC) 360 0 - (80 0 + 150°) 130 0 .
Bo BTOpOM CJIYLJae HMeeM: LAOe LBOC - LAOB 70° .
OmBem: 70 HJIH 130
40. B03MO)f{HblllBa CJ1Y4aH pacnOJIO)KeHH5I .nY4ei1 oe H OD BHYTPH
yrJ1a AOB:
1 ) Jlyq oe J1e>KHT BHYTPH yrJI3 AOD, CM. pHC. I;
2) Jlytl DC JIe>KHT BHYTPH yrJI3 BOD, CM. pHC. 2.
D
A
puc.2
pUC. 1 C
B
A
B nepSOM CJJYliae HMeeM: LAOB LAOe + LCOD + LBOD
50 0 + 40° + 80° 1 70° .
Bo BTOpOM CJ1Y 1 1ae HMeeM: LAOB LAOD + LBOD
LAOD + 80°, LAOD LADe - LCOD 50° - 40° - to°.
I1TaK, B 3TOM cJIYlJae LAOB 10° + 80° 90° .
OmBem: 90 HJ1H 170
47. TaK KaK JIYlJ OD J1e)KHT BHYTPH yrJIa AOO, TO LAOC + LBOD
180 0 + LOOD 3 1 0 0 , C M. pHC. I Ha CJ1e LlY lO weH CTp a HHlle.

OCHOBHbJe nOHHTHfi reOMeTpHH
119
LMOL 180 0 - LAOL - LBOM 180 0 - 155 0 25°, CM. pHc.2.
L
A
o
B
A
o
B
OmBem: 25
50. nYCTb x HCKOMaH rpaJ!ycHaS1 Mepa yrJIa BOD, Torlla rpaJlYc-
Hast Mepa yrJla AOe paSHa 4x. B03MO>KHbll!B3 cJlyqaSl pacnOJlO>KeHHSI
JlYQei1 OC H OD BHyrP" pa3BepHYToro yrna AOB:
I ) J1YlJ OC JIe)KHT BHyrpH yrJla AOD, CM. pHC. 1;
2) JlYll oe Jle)KHT BHYTPH yrJla BOD, CM. pHC. 2.
pRC. 1
pHc.2
A
o
D
B
A
o
B
B nepBOM CJ1YlJae HMeeM: LAOe + LCOD + LBOD LAOB,
OTKYlta CJIeJlyeT, liTO 4x + 50° + x 180 0 , 5x 130 0 , x 26°
(LAOB 180 0 - KaK pa3BepHyTblH yroJl, LeOD 50° - no YCJlO-
B"IO ).
Bo BTOpOM cJIYlIae HMeeM: LAOC + LBOD LAOB + LCOD,
T.e. x + 4x 180 0 + 50°, 5x 230 0 , x 46°. Ho eCJI" x 46°, TO
LAOe 4x 184 0 , liTO npOTHBOpeqHT YCJlOBHIO (yron AOe He MO-
>KeT 6blTb 60JlbWe 180 0 , T.K. o6a JIYlJa oe Ii OD no YCJJOBHIO Jle>KaT
BHyrP" OllHoro Ii Toro >Ke pa3BepHyTOrO yrJla). n03TOMY CJlYlJaH 2) Ha
C3MOM lleJIe He peaJlH3yeTCH.
Om8em: 26
59. ,l{JlSI B3aHMHoro pacnOJlO>KeHHSI JlYlJeH OA,OB H oe OTHOCH-
TeJIbHO Jlpyr Jlpyra HMelOTCH llBe B03MO>KHOCTH:
1 ) JIY4 OA JIe>KHT BHe yrJla BOe, CM. pHC. I;
2) JIyq 0 A J1e>KHT BHYTPH yr J1a BOC, CM. puc. 2.

120
PemeHHfl K 91
'"
B
B
pRC. 1
pUC. 2
A
A
B K3>K}lOM H3 CJ1YQaeB I) Ii 2) Jlyq 0 D MO>Ker 6blTb pacnonO)l{eJ-I
D.BYMH cnoco6aMH:
a) TaK, lJTO yro.n COD opHeHTHpoB3H no 4aCOBOH CTpeJIKe, CM. PH-
CYHKH la) H 2a);
B
B
pHc.la)
pHc.l6)
D
A
A
6) T3K, I.jTO yrOJ1 CO D opHeHTHpOBaH npOTHB tlaCOBOH CTpeJ1KH, CM.
PHCYHKH 16) H 26).
B
B
pRc.2a)
pHc.26)
D
A
o
A
OCYlUeCTBHM BblQHCJleHHe rpallycHoH Mepbl yrJla AOD B K3)1(llOM
..
H3 3THX qeTbIpex CJIyqaeB.
B c.nyttae I a) HMeeM: LAOD 360 0 - LADB - LBOe - LCOD
360 0 - 90° - 160 0 - 80° 30°.
B cJ1Yllae 16) HMeeM: LBOD LBOC LCOD 160 0 -80 0 80 0 ,
LAOD LAOB + LBOD 90 0 + 80 0 170 0 .

OCHOBHbJe nOHHTHfi reOMerpHH
121
B cJIY 1 lae2a) HMeeM: LAOC LBOC LAOB 160 0 90° 70°,
LAOD LCOD - LAOC 80 0 70° 10°.
B cJIyqae26) HMeeM: LAoe LBOC-LAOB 160 0 90° 70°,
LAOD LAoe + LCOD 70° + 80° 150 0 .
OmBem: 30 HJ1H 170 "JIM 10 HJlH 150
61. nYCTb Jyf cepellHHa OTpe3K3 AB, H - TotJKa nepeCeqeHHSI npH-
MOH AB c np5lMOH L. B03MO)KHbl D.B3 c.nY4afl pacnOJlO)KeHHH TOlJeK A
11 B OTHOCHTeJ1bHO npSlMoM l:
I ) T04Ka A pacn0J10)KeHa 6J1H)Ke K npSlMoH I, qeM TOI.JKa B, CM. pl1C. I;
2) TOllKa A paCnOJIO)KeH8 Ll3JIbWe OT np5lMOH l, qeM TO'-lKa B, CM. pHC. 2.
pac. 1
B
4M
A
10
pRc.2
10
H
1
H
1
B cnYllae 1) HMeeM: AH 10, AIH AH + AM
B CJJY l lae2) HMeeM: BH AH AB 10-4 6, AfH
6 + 2 8.
OmBem: 8 "JIM 12
10 + 2 12.
BH+BM
65. nYCTb!if cepeJJ.HH30Tpe3Ka AB, H - TOLJK3 nepeCeqeHHSI npSl-
MOM AB c npS1MOH l. B03MO>KHbl LlB3 CJIY1.JaH pacnOJlO>KeHHH T04eK A
H 1\;/ OTHOCHTeJIbHO npSlMoH l: I) T04KH A H M pacnOnO>KeHbl no OH-
HY CTOPOHY OT npSIMoH l, CM. pHC. 1; 2) T04KH A H M pacnOJIO>KeHbl no
pa3Hble CTOpOHbI OT npHMoH l, CM. pHC. 2.
pRC. 1
A
18 M
11
H
pRc.2
A
18
I
H I
11
M

122
B cJlYlJae I) HMeeM: AM AH - M H 18 - 11
2AM 14. B CJ1YlJae 2) HMeeM: AJ\;f AH + M H
AB 2AM 58.
OmBem: 14 "JIM 58
PemeHHH K  J
7, AB
18 + 11 - 29,
.,
69. nYCTb /vI H N OCHOBaHHSI nepneH!lHKYJISIpOB, npoBe,aeHHblx H3
T04KH A K npHMblM l 11 P COOTBeTCTBeHHO. TaK KaK npHMbIe l H P na-
paJlJleJ1bHbl, TO TOlJKH A, At!, N J1e>KaT Ha OllHOH npHMOH. 113 YCJlOBHH
TaK>Ke CJ1ellyeT, liTO AN 25, AI N 18. n03TOMY B03MO>KHbl TOJlb-
KO l{Ba cJlyqaH pacnOJlO>KeHHH TOlJeK A, M, N OTI-IOCHTeJlbHO Jlpyr Jtpy-
ra: I) TOllKa M J1e)KHT Me>KJ1Y TOllKaMM A H N, CM. PHCYHOK 1  2) TOllKa
N JIe)l{HT Me>KllY TOliKaMH A Ii M, CM. PHCYHOK 2. (TOT CJlY4aH, KO-
rlla TOIIK3 A J1e>KHT Me>KHY T04K3MH 1\1/ H N, HeB03MO)l{eH, nOCKOJ1bKY
AN > MN).
25
pac. 1 18
A M
N
pHc.2 25 18
e ..
A N M
B CJ1Y 1 1ae I) HMeeM: AM AN - J\;1 N 7. B CJ1YLJae 2) HMeeM:
AM AN + M N 43. TaKHM o6pa30M, paCCTOSlHHe OT TOllKH A 110
npHMOH l, paBHoe l]..nI1He oTpe3Ka AM, paBliO JIH60 7, JIH60 43.
OmBem: 7 HJ1H 43
75. 5YlleM C4HT3Tb, liTO npHM3S1 a pacnOJIO>KeHa BbIwe npHMOH b. ll.nSJ
u -
pacno.nO>KeHHS1 npHMOI1 C HMelOTCfI ,QBe B03MO>KHOCTH:
I ) npSlM8H C pacn0J10>KeHa HH)I(e npHMOH b, CM. pHC. 1;
2) npSlMaSi C pacnOJlO>KeH3 Bblllle npSlMoH b, CM. pHC. 2.
pac. 1 a
b
pHc.2 c
3
 
24
a
24
c
b
d 3
, , , ,
B I-OM cnYlJae paCCTOHHHe Me)KJlY npS1MblMM a H C pasHo 3 + 24 27,
a 80 2-0M CJlYlJae 24 3 21.

OCHOBHble nOHHTHH reOMerpHH
123
B Ka}l{JlOM H3 3THX ,QBYX CJlYlJaeB DJIH paCnOJIO>KeHHSI npS1MOH d T3K-
)l{e HMelOTCH .use B03MO)l{HOCTH: npHM351 d pacnOJlO)l{eHa HH)I{e npH-
MOH c, CM. PHCYHKH 1.1 H 2.1, HJlH npHM3H d pacnOJlO)l{eH3 BbIllJe npSl-
Mott c, CM. PHCYHKM 1.2 Ii 2.2.
pRc.I.1 a puc. 1.2 a d
  J   .
27 27 27
, , c , r , r
C J 
27
d , r
pRc.2.1 C pRc.2.2 d
 
21 27
a 27 c , r
 
d 21
a , r
B CJ1Y4ae 1.2 npSIMble a H d COBnaJlaIOT, 4TO npoTHBopetJHT YCJIO-
BHIO (no YCJIOBHIO Bee 4 npSIMbIe pa3J1H1.JHbl). 003TOMY 3TOT CJIYlIaH
HeB03MO>KeH.
A B CJJY l l3HX 1.1,2.1 H 2.2 J]JIH HCKOMoro paCCTOHHHSI h Me)l{llY npfl-
MblMH a H d nOJlyqae1 CJlen.YlOlllHe 3HalJeHH5I: h ' 27 + 27 54,
/t 27 - 21 6, h 27 + 21 48.
OmBem: 6 HJIH 48 HJlH 54
78. TaK K3K 5300 883 · 6 + 2, TO nOCaJlHB 883 .nepeB3 C HHTepBaJIOM
6 MeTpoB OAHO OT npyroro, Haq3B C 6 -fO Merpa HoporH, Mbl nOJlYQHM 1
LITO paCCTOflHHe OT nOCJle)!HerO !lepeBa ,LlO KOHlla JtoporH paBHo 2 MeT-
paM. CJleJlOBaTeJIbHO, 883 !tepesa, nOC3>KeHHbIX B OLlHH PSI)!, XB3TaeT,
4T06bI BblnOJIHHTb Tpe6yeMbie YCJIOBHH lIJ1H 3Toro pSll!a !lepeBbeB. A ec-
JIM OrpaHHlJHTbCH 882 -10 JlepeBbHMH, TO paCCTOHHHe Me)l{JlY KaKHMH-TO
u
llBYMSlllepeBbSIMH, HJ1H paCCTOHHHe OT KpaHHero epeBa ,no KOHLla J10pO-
rH, OKa)l{eTCSI 60JIbWe 6 MeTpoB. TaKHM o6pa30M, H8HMeHblUee QHCJIO
AepeBbeB, KOTopoe Tpe6yeTcSI nOC3JlHTb B Ka>Kl{OM PS1DY, P3BHO 883, a

124
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PeWeHHJ1 K J
Bcero Tpe6yercSI nocallHTb He MeHbllJe 883 · 6 5298 llepeBbeB.
OmBem: 5298
80. TaK KaK WHpHHa peKH paBHa 165 MerpaM, a K3)Kllb1H H3 ABYX KOH-
UOB MOCTa YllaJ1eH OT 6epera He MeHee, lleM Ha 30 MeTpos, TO MHHa
MOCTB He MeHee 165 + 30 · 2 225 MeTpoB. TaK KaK paCCTOstHHe II OT
JleBOrO KOHua MOCTa .QO 6JlH>KaHWeH onopbl He OOJ1ee 20 MeTpoB, a WH-
pHH3 C8MOH onopbl 1,5 Merpa, TO paCCTOHHHe OT J1eBOrO KOHua MOCTa
110 npaBoro KOHua 6nH.>KaHWeH K HeMY onopbI He OOJ1ee 21,5 MerpoB,
CM. PHCYHOK.
BaqaJIO MOCT8
I
I
I
.
I
I
I II' 20 M
onopy
/' '-
1,SM 1 2 ' 20M 1,5M
PaCCTOS1HHe OT npaBoro KOHa nepBOH onopbl !to npaBoro KOHua
BTOpOH onopbl T810Ke He 60Jlee 21,5 MerpoB. OTCIOJ!8 CJ1eJlyer, liTO eCJlH
MOCT HMeer 9 onop, TO paCCTOHHHe OT J1eBOrO KOI-lua Mocra 110 npaBoro
KpaH 9-0H onopbl He 60Jlee 21,5 · 9 193,5 MerpoB. npH 3TOM pac-
CTOSlHHe OT npaBoro KpaH JIeSHTOH OnOpblltO npaBoro Kpa51 MOCTa 6Y.ller
COCTaBJlHTb He MeHee 225 - 193,5 31,5 MeTpoB, liTO npOTHBopellHT
npeJl'bHBJlSleMbIM Tpe6oBaHH5IM. TaKHM o6pa30M, 9 onop Hen.OCT3TOQHO.
ECJ1H >Ke IIHC.nO onop B3HTb paBHblM 10, nepBYlo H3 I-IHX pacnoJlo-
>KHTb 1-13 paccToS1llHH 20 MeTpoB OT JleBOrO KpaH, a Bee nOCJleHYlOlllHe
lJepe3 20 MeTpoB OJlHa OT JlpyroH, TO npH llJIHI-Ie MOCTa 225 MeTpoB pac-
CTOSlHHe OT nOCJ1eJlHeH H3 HI1X (n.eCS1TOH 110 ClJeTY) ,no npaBoro Kpast MO-
CTa 6Yller paBHo 225 21,5 · 10 10 MeTpos - IlJIH TaKora Mocra Bee
rpe6yeMbie YCJlOBH5I 6YYT BbInOJIHeHbl.
Om8em: 10
83. nnocKocTb JleJIHTCSI lJeTblpbMH npHMblMH Ha HaHMeHbwee 803-
u
MO>KHOe lJHCnO lJaCTeH B TOM CJIyt.lae, eCJlH HH OJlH3 H3 npHMblX He nepe-

OCHOBHb/e nOHRTlfR reOMerpHl1
125
ceKaeT n-pyrylO, T.e. Bee 4 npHMble napaJ1JJeJlbHbl, CM. PHCYHOK I. 11 raK,
u
HaHMeHbwee lJHCJ10 llacreH, H3 KOTopble nJlOCKOCTb AeJlHTCSJ lJeTblpbMH
npSlMblMM, paBHo 5.
OmBem: 5
pHc.1
pHc.2
8
10
7
4
84. nJlOCKOCTb J1.eJII1TCSI IleTblpbMSI npHMblMH Ha H3H6oJlbWee B03MO)K-
u
HOC llHCJlO qaCTeli B TOM CJ1Y4ae, eC"TJH K3}1{!1ble JIBe H3 3THX npHMblX ne-
peceKalOTCH, CM. Pl1CYHOK 2 (06J18CTH, Ha KOTopble nJlOCKOCTb pa36HTa
npSlMblMH Ha 3TOM pHcYHKe, npoHYMepOBalibl). YlcKoMoe lIHCJ10 llacreCi
paBHo 11.
Om8em: 11
85. npenOJ)O)KHM, liTO Ha nJlOCKOCTH YlKe npOBeJ1eHO k npHMblX
u
a}, a2, . . . , ak, pa 3 11eJIHBIJJHX nJlOCKOCTb HB HeKOTopoe LJHCJlO llaCTeH, Ii
MbI llo6aBJlHeM elUe OJlHY npHMYIO - l. l(a)K!lbIH H3 3THX npOMe>KYTKOB,
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B
N
TaK K3K A1\,f B!vI H CM N lvI, TO COrJIaCHO 3a){a4e NQ9 no-
JIYllaeM, 4TO IleTblpexyrOJ1bHHK AN BC - napaJlJleJIOrpaMM (T.K. era
!lHarOI-laJ1H llCJ15ITCH TOIIKOH nepeCeIJeHHH nononaM). A nOCKOJIbKY
LAC B 90°, TO Bee yrJIbl napaJIJleJlOrpaMMa AN BC npSlMbIe, T.e.
01-( 51BJI5IerC5I np5lMoyroJlbHHKOM. TaK KaK JtHarOHaJII1 npSlMoyrOJIbHHKa
paBI-Ibl ( CM. 3ana l lY N214), TO C N AB. OCTaJ10Cb BCnOMHI1Tb, 4TO no
AB, CM
H Tpe6oB8J10Cb .QOKa33Tb.
16. I-Ia np5lMOH AI N OT T04KH N OTJIO)KHM oTpe30K N P, paBHbIH OT-
pe3KY 1\1 N, CM. Jt3HHbIH HHIKe PHCYHOK.
A
c
B

TpeyroJlbHHK
135
llHarOH3J1H LJeTbipexyroJlbHHK3 AMCP JleJISiTCH TOIIKOH nepecetle-
HHSf nOnOJIaM, n03TOMY AMCP napaJlJleJIOrpaMM (CM. 3BAaQy N9).
CJIe.aoB3TeJlbHO, AM CP, KaK npoTHBOnOJlO)KHble CTOpOHbl na-
paJlneJIOrpaMMa AMCP (CM. 3311aQy NQ7). A nOCKOJIbKY BM AM,
TO BM CP. OCTa.nOCb 33MeTHTb, LITO nOCKOJIbKY npOTHBOnOJIO>K-
Hble CTOpOHbl BM H CP lleTbIpexyroJlbHHKa BMPC paBHbl H napaJl-
JIeJJbHbl (npHMaH CP napanJleJIbHa npHMOH AB, T.K. AMCP - na-
paJJJ1eJ10rpaMM), TO BM PC HBJlHeTCSI napaJIJleJIOrpaMMOM (CM. 3aJlB-
try NQ 1 0 ). TaKH M o 6 pa30M, oTpe30K M N napaJlJIeJleH cTopoHe BC H
17. nYCTb E - TOLJKa nepeCeqeHHSI npSlMblX BN It AD, CM. PHCY-
HOK. Tor,aa TpeyroJlbHHKH BCN H EDN paBHbl no BTOPOMY npH3HaKY
(LCBN LNED - KaK HaKpecT Jle)KalllHe yrJlbl np" napaJlJleJIbHblX
npS1MbIX, LBNC LEND KaK BepTHKaJIbHbie yrJlbI, CN DN
no onpeJleJIeHHIO TOlJKH N). 113 paBeHCTBa TpeyroJlbHHKOB BCN "
EDN CJIeJlyeT, liTO DE BC H BN N E.
B C
A D E
TaKKaK M H N cepen.HHbIoTpe3KoB AB M BE, TO MN-cpeJ1-
HS1 Jl H HMSI T p e yr oJlb H HKa ABE. n03TOMY O Tp e 30 K M N napaJJJleJleH
Tpe6oB3J10Cb )!OKa3aTb.
27. nYCTb a, /3, 'Y rpanycH3S1 Mepa BHyTpeHHMX yrJloB "PH BepwHHax
A, B, C COOTBeTCTBeHHO. TOfJ13 rpan.ycHaSi Mepa BHelllHHX yrJIOB np"
BepWHHax B H C paBHbI 180 0 /3 H 180 0 'Y. 113 YCJlOBHH HMeeM:
(} + 180 0 /3 + 180 0 'Y 196 0 . flpeo6pa3yeM JleBYIO lJaCTb 3Toro
p3BeHcTBa, npH63BHB K Heft H OTHSIB a, a 3aTeM crpynnHpoBaB CJlarae-

136
MbIe CJle.nyI01UHM 06pa30M: a + 180 0 - 13 + 180 0
2a + 360 0 - (Q + ,8 + 1') 20 + 180 0 .
I1TaK, HaMH nOJlyqeHO paBeHcTBo 20 + 180 0
HaxonHM, I:fTO a 8° .
OmBem: 8
PemeHHfI K 3
,+a-Q
.
196 0 , H3 KOToporo
152 0 , CM. OPMYJ]Y
H3 3a.llaIJH N252.
113 33113 1 1" NQ53.
52. nYCTb P - TOlJKa nepeCelJeHHH .nytla GO co
CTOpOHOH AB, CM. PHCYHOK. TaK KaK yr.nbl AO?
Ii BOP - BHeWHHe yrJIbl TpeyrOJIbHHKOB ADG H
BOG cooTBeTcTBeHHo, TO
LAOP
LBOP
A P,
+LC)
58. nYCTb ABC - npSlMoyro.nbHblH Tpe-
yroJlbHHK c rHnoTeHY30H AB, B KOTOpOM
LB 48°, a oTpe3KH G!\t/ H C H COOT-
BeTCTBeHHO MellH3Ha H Bblcor3 K rHnoTeHY3e, A
CM. PHCYHOK. TorJla
LA 90° - LB 42°, LBCH 90° - LCBH 42°.
TaK KaK MeHaHa K rHnoTeHY3e npHMoyronbHoro TpeyrobHHKa
pasHa nOJIOBHHe I"'HnOTeHY3bl (CM. 3allalJY NQ 15), TO C 1\;J AM. no-
3TOMY LG AM LAG M - KaK yr JIb) npH OCHOBaHHH paBH06eJlpeHHO-
ro TpeyrOJ1bHHKa. TaKHM 06pa30M HMeeM: LACM 42°, LBG H 42°,
oTKyn.a LlvICH 90 0 - LAGM - LBCH 6°.
OmBem: 6
48°
M H B

Tpeyro.IIbHHK
137
62. nYCTb ABC - npHMoyroJlbHbIH TpeyronbHHK C rHnoTeHY30H AB,
a oTpe3KH C H H C M era BbICOT3 H MeJ!HaHa. 6e3 OrpaHHlJeHMH o6IQ-
HOCT" MO)l{HO ClIHTaTb, liTO LA MeHbWHH H3 OCTpblX yrJIOB 3Toro Tpe-
yrOJlbHHK3, T.e. LA  45°. Torlla, nOBTopSlS1 BblKJIallKIt, npHBelleHHble B
peweHHH 38Jl3 1 lH N258, nOJIYlIHM, 4TO LAC M LA M LBG H LA.
nYCTb C L 6HcceKTpHca yrJI3 AC B, CM. 113HHbIM HH)I{e PHCYHOK,
TOf1l3 LAG L LBC L 45°. OTclOlla nOJlY[laeM, lJTO LM C L
45° LACM 45° LA H LHCL 45° LBCH 45°- LA,
T.e. L}.;[CL LHCL, LITO H Tpe6oBaJIOCb llOKa3aTb.
C
1
I
I
I
I
I
I
A M LH
64. TaK KaK AP H BP oHcceKTpHcb l y rJI OB LA M L B, CYMMa KOTO-
ABP npHMoyrOJIbHblH TpeyrOJIbHHK C rlnoTeHY30H AR. AHaJIO-
rHQHO, CDQ npHMoyroJlbHblH rpeyroJlbHHK C rHnoTeHY30H CD.
nYCTb M H N cepeAHHbI 60KOBbiX CTOpOH AB 11 CD. TaK K3K
MellHaH3 K rHnoTeHY3e npHMoyrOJIbHOrO TpeyrOJ1bHHKa pasHa nOJIOBHHe
rHnOTeHY3bI, TO P]\;f A}.;! Ii, 3HalJHT, Ll\l P A Llvf AP, (KaK yr-
JIbl npH OCHOBaHHH paBH06el1peHHOro Tpeyr0J1bHHK3), CM. ltaHHblH HH-
lKe PHCYHOK I. Ho LM AP LP AD (T.K. AP 6HcceKTpHca LA),
n03TOMY LM P A LP AD. OTCIOH3 CJIeJlyeT, QTO npS1M351 M P na-
paJlJIeJIbHa npHMoH AD.
pHC. 1
B
c
pRc.2
B
c
A
D
A
D

PeweHHH K 3
OTpe30K M N S1BJlSleTCH CpeJlHeH nHHHeH Tpane UHH, n03TOMY OH
3aMeTHM, liTO nOCKOJIbKY M P II AD H M N II AD, TO npSlMble M P H
M N COBnaJlalOT (T.K. 113 TOIJKH M MO>KHO npOBeCTI1 JIHlllb OD.HY npSl-
MYIO, napaJlJIeJlbHYIO AD). CneJlOB3TeJlbHO, TOlfKa P JIe>KHT H3 npHMoH
M N. A6cOJllOTHO aHanOrHLJHO llOKa3bIBaeTCSI, liTO TOqKa Q Jle>KHT H3
npSIMOH M N.
TaK KaK o6e T01.JKH P H Q Jle)l{3T H3 npHMOH M N, TO B CJIyqae,
11 3 06 pa)l{eHHOM H 3 pH C. I, n OJI YlJaeM: PQ !vI N M P - NQ
liTO M N > M P + N Q, 4TO B CBOIO 04epe.nb BblnOJIHSIerCSI TOr,aa, KO-
rA32MN AD + BC > AB + CD 2(MP + NQ). ECJlH >Ke
AD + BC < AB + CD, TO MN < MP + NQ, n03TOMY HMeeT Me-
CTO CJlyqaH, H300pa>KeHHb1H H3 pUC. 2 (T04Ka Q J1e>KHT 6JJH>Ke K TOLlKe
M, qe M TOqK3 P). B 3TOM CJIYlJ ae MP + N Q MN + PQ, oTKy,na
138
eCJIIi AD + BC > AB + CD, JlH60 PQ
eCJlH AD + BC < AB + CD, T.e. PQ
liTO H Tpe6oBaJIOCb .llOK33aTb.
66. I) 4epe3 T04KY C npOBeneM npHMYIO, napaJIJIeJlbHYIO CTOpOHe
AB, H TOlJKY nepeCe4eHHH 3TOH npHMOH C npHMOH AD 0603HalJHM 4e-
pe3 E, CM. PHCYHOK 1. TaK KaK LBAD LCED H no YCJlOBHIO
LBAD + LCDA 90°, TO LCED + LCDA 90°, T.e. CDE -
npHMoyroJlbHblH TpeyrOJIbHHK C rHnoTeHY30M DE.
pHc.l
B
c
pHc.2
B K C
A
E
D
A
ELM D
nYCTb K H L cepe.nHHbl oCHoBaHHH BC H AD. no YCJlOBHIO

TpeyroJlbHHK
139
HaM Tpe6yeTcSI JlOK333Tb, liTO K L
CKOJlbKY DE AD - AE AD BC (AE BC, K3K npoTHBono-
JlO>KHble CTOpOHbl napaJ1J1eJ10rpaMMa ABC E), TO HaM Tpe6yeTcS1 .llOK3-
qTO K L C M, CM. pHC. 2.
2) 4T06bl JlOKa3aTb paBeHcTBo KL CM, nOK3)1(eM, lITO qeTbI-
pexyrOJIbHMK KCM L napaJIJIeJIOrpaMM.llJlH 3Toro H3M !lOCT3TOllHO
nOK333Tb, liTO KG LM (Torlla KCM L 6YJIeT napaJIJleJlorpaMMoM
no npH3H3KY «npoTHBOnOJIO>KHble CTOpOHbI napaJlJleJlbHbl H paBHbI»).
67. nYCTb LN MeAHaH3 npHMoyroJlbHoro TpeyrOJlbHHKa C LI(, CM.
u
Jl3HHbIH HH>Ke PHCYHOK.
K A N C

140
PellleHHH K 3
III'
n03TOMY J{OCT3TOlJHO J10Ka3aTb, liTO LN AL.
3aMeTHM, liTO nOCKOJlbKY LCLN - LNCL (KaK yrJlbl npH OCHO-
BaHIi" paBHo6e.npeHHoro rpeyroJlbHliKa CLN), a LNCL LBCL
(T.K. CL - 6HcceKTpHca yrJla C), TO LCLN LBCL. n03TOMY npSl-
MaSl LN napaJIJ1eJ1bH3 npSlMoH BC.
113 napaJJneJIbHOCTH npSlMblX LN Ii BC CJlen.yeT, liTO LANL
- LACB. OCTanOCb JlHWb 3aMeTHTb, IjTO nOCKOJlbKY LACB LCAB
(KaK yrJJbl npH OCHOBaHHH paBHo6e-npeHHoro TpeyroJlbHHK3 ABC), TO
LANL LCAB LNAL. TaK K3K B rpeyroJlbHHKe ALN yrJ1bl npH
BepwHH3x A H N paBHbl, TO OH HBJlSleTCH paBH06eJlpeHHblM C OCHOBa-
HHeM AN, T.e. AL LN, liTO H Tpe6oBaJlOCb llOKa33Tb.
72. 0603H3lJJtM tlepe3 a H (3 rpallYCHbIe Mepbl yrJIOB A H B TpeyroJlb-
HHKa ABC.
a) ECJlH MeDJ13H3 Clv! paBHa AB /2, TO CM AM - BM. Torlla
H3 paBHo6ellpeHHocTM rpeyroJlbHHKOB AClv[ 11 BClvI CJleD-yeT, liTO
LACk! LCAAI = a, LBCM LCBM (3, CM. pHC. 1. OrclOlla
nOJlY4aeM: LACB LACM + LBCM Q + {3. OCTaeTcH JlHUJb 33-
MeTHTb, liTO LACB + LA + LB (Q + (3) + a + {3 180 0 (KaK cYMMa
yrJloB TpeyroJibliHKa ABC). n03TOMY Q+ (3 90°, liTO Ii Tpe6oBaJlOCb
llOKa3aTb.
M
puC. 1
A
M
pRc.2
B
c
c
A
M
puc.3
B
6) nYCTb CM < AB /2 AM BM, CM. pHC. 2. TaK KaK B Tpe-
u u
yroJlbHI1Ke npoTHB MeHbweH CTOpOHbl Jle)f(HT MelibUJHH yroJl, TO 113 Tpe-
yronbHHKOB ACM H BCM HMeeM: LACM > 0:, LBCM > {3. CJlO-
>KHB 3TH J1Ba HepaBeHCTBa, nOJlYIIHM: LACB LACM + LBCM >
> a + {3. OTKYLla CJI, eJ!yeT 4TO 2LACB > a: + (3 + LACB 180°,
LACB > 90°, iJTO H Tpe6oB3JIOCb J{OK333Tb.
B) EcnH CM > AB /2 AM BM, CM. pHC. 3, TO 3HanOrHQHO

TpeyroJlbHHK
nYHKTY 6) HMeeM: LACB
2LACB < a + {3 + LACB
.llOKa3aTb.
141
LACM + LBCM < a + {3, oTKYJ1a
180 0 , LACB < 90°, liTO H Tpe6oBaJlOCb
73. nYCTb ABC TpeyrOJlbHHK, B KOTOpOM AB > AC, AM Me,nHa-
Ha 3Toro TpeyroJlbHHKa. TorHa, no YCJlOBHIO, HaM Tpe6yeTcH ,nOKa3aTb,
liTO yroJi BAlvl MeHbwe yrJI3 CAM. Ha npOJlOJl)l{eHHH MeJ!HaHbI AAI
33 T01IKY J\;[ OTJlO>KHM OTpe30K MI(, paBHblH oTpe3KY AM, CM. PHCY-
HOK.
A
B
c
K
TpeyronbHHKH ACM 11 BKM paBHbl no nepBoMY rlpH3H3KY
(BAI CM, T.K. AM MellHaHa, MK A1\tf - no nocTpoeHlilO,
LBMK LAMC K8K BepTMK3J1bHble yrJlbl). OTCI0118 CJleJ1yeT, liTO
BK AC 11 LCAM LBK }vI.
TaK KaK B TpeyrOJIbHHKe HanpoTHB MeHbweH CTOpOHbl JIe)KHT MeHb-
ill"H yroJl H BI( < AB (BK AC < AB, no YCJlOBHIO), TO yrOJI BAK
MeHbwe yrJIa BKA. OCT3JIOCb JlHlllb 33MeTHTb, I)TO LBAK LBAM
11 LBKA LBKM LCAM, T.e. LBAM < LCAM, liTO H Tpe-
60B3JIOCb 1!OK33aTb.
74. a) Ha npOllOJl>KeHM" MellH3Hbi Blvl 3a TOttI<)' M
OTJIO)KHM OTpe30K MD, paBHblM OTpe3KY BM, CM. PH-
CYHOK. TaK KaK )lHarOHaJlH IJeTbipexyrOJIbHHK3 ABCD
u
JleJISlTCSI TOtJKOH nepeCeqeHHH nononaM, TO 0" SlBJlHeT-
CH napanJleJIOrpaMMOM. n03TOMY CD AB.
B
A
D
HaJlee, no HepaBeHcTBY TpeyrOJlbHHK3 MMeeM: BD < BC + CD.
nOJlCTaBJISlSI B 3TO HepaBeHcTBo BD 2BM Ii CD AB, nonYlJaeM:

142 PemeHHH K 3
6) 113 6ABNI H 6CBM COrJlaCHO HepaBeHCTBY TpeyroJlbHHK3
HMeeM: BM + AM > AB, Blv! + Clv! > BC. CJIO>KHB 3TH llB3 He-
paBeHCTB3, n0J1Y 1 IHM: 2B1\1 +AM +C M > AB+BC , OTKYJt3 CJlen.yeT,
npUMettaHUe. B HOKa3aTeJ1bCTBe HepaBeHCTB3 nYHKTa 6) MbI HHKaK He
HCnOJlb30BaJ1H TO 06CTOHTeJIbCTBO, LITO M cepellHHa OTpe3Ka AC.
CJIellOS3TeJIbHO, llaHHoe HepaBeHCTBO CnpaBeJUIHBO ,QJIS1 JlI06oro OT-
pe3K3 B/..1, f!le /-.;[ - npOH3BOJlbHaSl BHyrpeHHfl5I TOIJKa CTOpOHbl AC.
75. nYCTb 1\;[ npOH3B0J1bHaS1 TOtlKa nJIOCKOCTH, OTJIHlJH3H OT BepllJHH
npSIMoyroJlbHHKa ABCD, CM. pI1CYHOK.
B C
"-
", ,." M
" ."
,.,

,
, ,
, ,
, ,
......
A
ilOKa>KeM, liTO AM < BM + CM + DAl. 113 TpeyroJlbHI1KOB AClvI
H BDM no l-iepaBeHcTBY TpeyrOJIbl-IHK3 HMeeM: A1\;l < AC + Ck!,
BD < BAlf + DA1. A nOCKOJ1bKY AC BD, KaK D.HarOH8J1H npHMO-
yrOJIbHHKa, TO 3aMeHH5I B HepaBeHCTse AlVl < AC + C/ttI BeJIH1IHHY AC
Ha BeJlHllHHY BD, I10JIYllaeM: ANI < BD + C /vI < (BM + DM) + C1\;1,
H pacKpblB3H cKo6KH, n0J1Y 1 1aeM Tpe6yeMoe l-IepaBeHcTBo.
lloKa33TeJIbCTBO HepaBeHcTB3 A1\;[ < BJ\t/ + CM + DM B TOM CJIY-
lJae, KOrJla T04Ka M COBna!laeT C OD.HOH 113 BepWHH npHMoyrOJlbHHKa,
..
OTJIHIJaeTCSI OT npHBeJleHHOrO Bblwe JlHWb TeM, lJTO OllMH 113 TpeyrO.flb-
HHKOB ACM MJIH BDM BblpO)K)laeTC5I B orpe30K, If, COOTseTCTBeHHo,
OJlHO H3 !lBYX HepaBeHcTB AM < AC + Clvf "JIM BD < BM + DM 06-
paaeTcH B paBeHcTBo. TaKHM o6pa30M, Tpe6yeMoe B 3a)lalJe HepaBeH-
CTBO J{OK83aHO HaMH l!J]H Bcex B03MO)KHbIX BapHaHToB pacnOJIO}l{eHHSI
3aJlaHHOM TOlJKH M OTHOCHTeJlbHO BepwHH npS1MoyrOJIbHHKa ABCD.
76. a) nYCTb K - TOlfKa nepeCelJeHHH npHMoH BM co CTOPOHOH AC,
u
CM. n.3HHbIH HH>Ke PHCYHOK.

TpeyrOJ1bHHK
143
c
A
Tor.na no HepaBeHCTBY TpeyrOJ1bHMKa AM < AK + K M. npH6aBHB K
06eHM qaCTHM 3Toro HepaBeHcTB3 JlJIHHY BM, nonYLJHM: A1\f + BAt! <
< AK + KM + BM. OTClOl1a, 3aMeHS1H CYMMY KM + BM JlJ1HHOH
OTpe3K3 BK, nOJIYllaeM, 4TO AJ\;/ + BM < AK + BK. A nOCKOJIbKY
BI( < CK + BC (cOrJI3CHO HepaBeHCTBY TpeyroJlbHMKa H3 6BKC),
TO AM +BM < AK +CK +BC. Y4HTbIBafl, liTO AK + CK AC,
nOJlY4aeM Tpe6yeMoe HepaBeHcTBo: AM + BM < AC + BC.
6) 4 To6bl ,llOKa3aTb HepaBeHCTBO nYHKTa 6), 1l0CTaT04HO 3aMeTHTb,
LITO BMeCTe C HepaBeHCTBOM nYHKTB a) CnpaBeJlJ1HBbl elUe !lBa HepaBeH-
CTsa, nOJ1HOCTblO aHaJIOfH1.JHbIX eMY:
AM+CM<AB+CB H CM+BM<CA+BA.
06a YKa3aHHbiX BblllJe HepaBeHCTBa nOJlYlJ31OTCH H3 HepaBeHCTBa nYHK-
Ta a) np" nepeCT3HOBKe 6YKB, 0603H3LJalOlUHX BepllJHHbl (nepBoe "3
HepaBeHCTB nOJlYLJaeTCH nepeCT3HOBKOH npyr C JlpyroM 6YKB B H C,
a BTopoe nepeCTaHOBKOH 6YKB A H C). CJlO>KHB JleBYIO H npaBYIO
43CTb HepaBeHcTBa AM + BM < AC + BC COOTBeTCTBeHHO C JIeSOH H
npasoH ll3CTbIO YKa3aHHbix BbIllle HepaBeHcTB AM + C M < AB + BC
M BM +CM < AB+AC, Mbl nOJIYlIHM, liTO 2AM +2BM +2CM <
< 2AB + 2BC + 2AC. A nOCJJe COKpallleHHH 06eHX qaCTeH nOJIyqeH-
Horo HepaBeHCTBa H3 2, nOJlYLJaeM Tpe6yeMoe HepaBeHcTBo.
77. CH343J1a ,llOKa}KeM, lITO lIJIH BblnYKJlOrO lJeTbIpexyrOJIbHHKa ABC D
CYMM3 LLJIHH llHarOI1aJIeH 60JlbWe, qeM ero nOJlynepHMeTp. 0603HallHM
lJepe3 P TOLJKY nepeCeqeHHH llHaroHaneH AC H BD, CM. JlaHHblH HH}Ke
PHCYHOK, H MH Ka>KJlOrO 113 TpeyrOJIbHHKOB ABP, BCP, C DP 11 ADP
BOCnOJIb3yeMcS1 HepaBeHcTBoM TpeyroRbHHK3: AB < AP + B P,
BC < BP + CP, CD < CP + DP, AD < DP + AP. CJIO}KHB
nOlJJIeHHO 9TH lIeTblpe HepaBeHCTBa, nOJlY1.fHM:

144 PeweHlfH K 93
AB+BC+CD+AD < 2(AP+BP+CP+DP) 2(AC+BD).
c
A D
Tenepb ltOKa)KeM, LITO CYMMa JLTIHH llHarOHaJleH AC H BD MeHbWe,
lJeM nepHMeTp tIeTblpexyrOJIbHHKa ABCD. 5e3 OrpaHHlJeHI151 06111HO-
CTM MO)KHO ClJHT3Tb, lJTO BC  AD, KaK B CJIYllae, H306pa)KelHOM H3
Jl8HHOM BblllJe pHcYHKe (eCJ1H BblnOJ1HeHO npoTHBOnOJlO>KHOe HepaBeH-
CTBO, T.e. BC  AD, TO MO)KHO npOH3BeCTH nepe0603HalJeHHe seplllHH
lJeTblpexyroJlbHHKa). 113 TpeyrOJIbHHKOB ABC H BCD no HepaseH-
cTBYTpeyro.nbHHKa iMeeM: AC<AB+BC H BD<BC+CD. CKJIa-
llblB3H nOllJIeHHO 3TH D.sa HepaBeHCTBa H YllHTbIBaSi TO, liTO BC  AD,
nOJIYlJaeM: AC + B D < AB + BC + BC + CD  AB + BC + AD+ CD,
liTO" Tpe6oB3J10Cb }lOK333Tb.
78. 1) OYCTb ABCD E npOH3BOJlbHblH BblnYKJlblH nSITHyrOJIbHHK,
aI, a2, · · · , as - Jl)IHHbI era CTOpOH AB, BC, · · · , AE, a d}, d 2 , · · · , d 5
J]JIHHbI ero J{HarOHaJ1eH AC, BD, . . . , BE, CM. PHCYHOK I.
B
A
c
pRC. 1
Q]
E
a4
Tor.n3 COrJIaCHO HepaBeHcTBY TpeyrOJIbHHKa, 3anHcaHHoMY JlJIH Tpe-
yronbHHKOB ABC, BCD, CDE, DEA H ABE, HMeeM: d 1 < al + a2,
d2 < a2 + a3, d3 < a3 + a4, d 4 < a4 + a5, d 5 < al + as. HeCJIO)KHO
BHD-eTb, liTO Ka>KllaSl H3 CTOPOH nHTHyrOJIbHHKa yqaCTByeT POBHO B D.BYX

TpeyrOJlbHHK 145
H3 3THX nHTH HepaBeHCTB. 003TOMY, CJlO)KHB n01.JJleHHO Bce 3TH Hepa-
BeHCTBa, nOJlYlJHM: d 1 + d2 + d3 + d4 + d s < 2(al + a2 + a3 + a4 + as),
u
T.e. CYMM3 J])1MH JlMarOH8J1eH nSiTHyrOJIbHMK3 MeHbwe ero YJlBOeHHOro
nepHMeTpa
2) OYCTb [{ L[ NP - nS1THyrOJIbHHK, sepwHHaMH KOToporo HB-
JlHIOTCS1 TOQKH nepece4eHHH JlHarOHaJleH nS1THyroJlbHI1Ka ABCDE, CM.
PHCYHOK 2.
B
A
c
pRc.2
Q]
E
a4
TorJt3 3anHC8B HepaBeHcTBo TpeyrOJIbHHK3 MH TpeyrOJIbHHKOB
ABP, BCK, CDL, DEM H AEN, nOJlYlIHM: a} < AP + BP,
a2 < BK +CK, a3 < CL+DL, a4 < DM + EM, a5 < AN + EN.
CJIO)l{HB 3TH 5 HepaBeHCTB, nOJIYlIHM, liTO a} + a2 + a3 + a4 + a5 < I,
rlle l- nepHMeTp 3aMKHyroH n.ecSlTH3BeHHoH JIOMaHOH (nHTHKOHeq-
HOft 3Be3Jlbl), KOTopaH BblD.eJIeH3 H3 pHc.2 )l{HpHOH JlHHHeH. OCTaeTcH
JlHWb 33MeTHTb, liTO nOCKOJIbKY YK333HHaH JlOM3Ha51 MO>KeT 6blTb npe.n-
CTaBJIeHa KaK 06'be.nHHeHHe Bcex nSlT" llHarOHaJleH AC, BD, CE, DA
H BE, 113 KOToporo Y!lanHJlM CTOpOHbl nHTHyrOJ1bHHKa K LM NP, TO
l < d 1 + d 2 + d 3 + d 4 + d s .
TaKHM 06pa30M, al + a2+ a3+ a4 + a5 < l < d 1 + d 2 + d 3 + d 4 + d 5 ,
liTO 33BeprnaeT Tpe6yeMoe .nOKa33TeJlbCTBO.

146
PellleHHR K 94
PemeHHH K 4 (nOJl06Hble TpeyroJlbHHKH)
HOMepa 3aiJatt, K KomopblM npuBeiJeflbl peUleHUR: 12, 13, 15,
17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30.
12. B) TaK KaK AP : CP < 1, TO T04Ka P Jle>KHT 6J1H>Ke K BepwHHe A,
lJeM CepeD.HHa oTpe3Ka AC. n03TOMY 0603HalJHB lJepe3 0 TO'-lKY nepe-
CelJeHHH HHarOH3J1eH napanJ1eJ10rpaMMa ABCD, nOJIYlJHM. liTO T01.JKa
P ne)l{HT H3 OTpe3Ke AO, npH 3TOM npHMaH BP nepeCeKaeT npHMYIO
CD B TOIJKe Q, J1e>KallleH Ha np0.llOJI)KeHHH CTOpOHbl CD 3a TOLJKY D,
CM. PHCYHOK Ha CJJe,nYlOlllet't CTpaHHue.
Yl3 nOllo6HSI TpeyrO.flbHHKOB ABP H CQP CJIe.uyer, liTO
l CD
(AB CD KaK npoTHBOnOJIO>KHble CTOpOHbl napanJleJ10rpaMMa).
.
oTKyn.a HaXOlXHM, liTO CQ : DQ
OmBem: 1 : (1 - p)
CD
CD
p
1 CD - CD
p
. (l-p)CD
.
p
l-E. CD
p ,
1
1- p.
D.anee HMeeM: DQ CQ
13. r) TaK KaK AD: DK > 1, TO DK < AD. n03TOMY B03MO}l{Hbi
CJlellYIOlllHe llB3 cJlyqaSl pacnOJIO>KeHHH TOIJKH K OTHOCHTeJ1bHO
TOqeK A H D:
I) TOqKa K JJe)KHT H3 Jlyqe AD 38 TOtIKOH D, CM. PHCYHOK I;
2) TOqK3 K JIe>KHT H3 oTpe3Ke AD, CM. PHCYHOK 2.
B 060HX CJIY4aHX 113 nOJl06HH TpeyrOJIbHHKOB APK H CPB CJIellyeT,

nOJ1.06Hble TfJtY'"().II IJ/IIII(11
liTO AP AI -- A.\.. (JJ(-Y
CP [](} -. l1/) 
pOHbI napaJIJll"JI()I'I);IMMcl).
pHc.l
147
AD - K3K npOTHBOnOJlO>KHble CTO-
puc.2
A D K A K D
B C JI Y(I('l t li: ()()I)(I )I {el ll IOM Ha pHC . 1 , HMeeM : AK AD + D K. TaK
.
[3 CJIY1IcIC, JI300pa)f{el-iHOM Ha pHC. 2, HMeeM: AK AD - D K
AD - 1 AD k - 1 AD oTK Y JIa AP AK k - 1
k k ' OP AD k.
15. nYCT1) P TOIIK3 11CpeCelJeHHH BblCOTbl AH co CTOpOHOH KBa.n.pa-
.ra [( L, CM. JlalIIbli11IHiI{C PHCYI-IOK, a x - HCKOMaH CTOpOH3 KBaJlpaTa.
'lol'Jla PH [(N x, AP AH - PH h x.
A
p
L
x x
B
N
HMC
KL
( OTHOLUeHHe cooTBeTcTBYIOUJ}fX CTOpOH paBHo OTHOlUeHHIO BbICOT, npo-
BeLlel-lHblX K 3TI1M cTopoHaM). nOJlCTaBJ1HSI B 3TO paBeHcTBo K L x,
BC a, AH h, AP h x, nOJlyqaeM CJIeJ1YlOlllee ypaBHeHHe

148
PellleHHH K 9 4
DJIH HaXO>KD.eHHH x: x
x h..
D N E.. cJlellyeT lITO AA-J + B 1\,1
CN q , BA1
_ D N + C N P + '1. T e AB C D P + fl
CN q' · · BM CN q.
nYCTb P - TOllKa nepece1leHHH npSlMbIx AB Ii CD, cM.l1aHHlJIH HI1-
>Ke PHCYJ-IOK. TaK KaK npHMble AD H BC napaJ1J1eJIbHbl, TO rpeyroJlb-
AP D P AD a
HHKH APD Ii BPC n Ollo6Hbl, n 0 3TOMY BP C P BC b ·
,
A8 CD a-b
BP C P b.
D
CD a - b H3 co-
CP b
CD P + fl
q t
nOJlYLJHM: 8M ON a - b . E + [ (a - b)q
BP C P b. q b(p + q) ·
n03TOMY BM+BP CN+CP (a-b)q+b(p+q) Te
B PCP b (p + q) , · ·
PM PN a +  0
BP CP b p + q) · CT3J10Cb JlHWb 38MeTHTb, liTO nOCKOJ1bKY
P BC (no BTOPOMY npH3HaKY nOllo6HSI) TO }vI N P /vI aq + 
, B C BP b(p+q).
T MN BC a + a g + b2.
13KHM o6pa30M, · b + q) p + q ·
OmBem: aq + bE-
p+q
A

nOJJ.06Hble TpeyroJlbHHKH
149
18. nYCTb AM : BM p. TaK K3K oTpe30K M N napaJlJIeJleH OCHO-
BaHHSlM AD 11 BC, TO no TeopeMe 0 npOnOpUJ10HaJlbHblX OTpe3K3X
(,apyroe H33BaHHe 0600meHHaSl TeopeM3 <l>aJleca) nOJlYlJaeM, liTO
AM : BM DN : CN p: 1. 113 3THX paBeHCTB COrJlaCHO pe3YJlbTa-
a b.
a b.
.. a 1
Toporo nOJ1Y1.la eM, li TO p lb.
MN
19. a) npOBeJleM Qepe3 TOqKY B 1 npHMYIO, napaJIJleJlbHYIO CC 1 , Ii
TOlJKY nepecetJeHHS1 3TOH npHMoH co CTOpOHOH AB 0603HatJHM LJepe3
K, CM. JlaHHblH HH}Ke PHCYHOK. no TeopeMe 0 n pOnOpUJiOH3J1 b Hb I X OT-
A
4
5
B
c
nYCTb AK 4x, TOr)l3 KC t 5x, ACt AK +KGl 9x. TaK
2 2
KaK no YCJIOBHIO AGl: GIB 3: 2, TO GIB 3 AC l 3 . 9x 6x.
OcraeTcSI JlHWb 3aMeTHTb, liTO no reopeMe 0 nponopUHOHaJlbHbIX
OTpe3K3X BP Bel 6x 6 .
PEl GIK 5x 5
Om8em: 6: 5
21. 3 aMe T HM , lITO paBeHCTBO BK 2 KL · KM paOHOCHJlbHO paBeH-
TpeyroJlbHHK BKC nOllo6eH TpeyroJlbHHKY MKA no JJ.BYM paB-
HblM yrnaM: LECK LMAK K3K HaKpeCT Jle>KalUHe YfJlbl npH na-

150
PellleHHH K  4
-
paJlJIeJlbHbIX npHMbIx BC, AM H ceKYUleH AC, LAK M LCKB
K3K BepTHKaJIbHbIe yr JIbI, CM. JlaHHbIH HH)I(e PHC Y H O K. 11 3 nOllo6HH Tpe-
B C
ADM
OCTaeTcS1 JIHlllb 3aMeTHTb, liTO nOCKOJIbKY npHMbIe AB H C L na-
paJIJleJlbHbI, TO TpeyrOJIbHHKH CLK H ABK T3K>Ke nOJl06HbI 11, 3Ha-
l CK KL
IHT, AK BK ·
CK H CK K L CJIell y eT lITO
AK AK BK '
BK K L {:} Bl(2 KL · [(M, lITO Ii T p e6oBaJ10Cb JlOKa3aTb.
K M BI(
22. nYCTb LA Q, Tor1l8 H3 npHMoyrOJIbHblX TpeyrOJIbHHKOB AC H
H ABC HMeeM: LAC H 90° - Q H LABC 90° - a, T.e.
LAC H LABC, CM. Jl3HHblH HH/Ke PHCYHOK.
C
A H B
a) TaK KaK TpeyroJlbHHKH CAH 11 BCH nOJl06HbI no JlBYM yrJlaM
(LA CH L CBH 90° a, LAHC LBHC 90°), TO
AH.BH CH 2 .
6) TpeyrOJIbHHK AC H nOllo6eH TpeyrOJIbHHKY ABC no HBYM yr-
AC AH 2
JIaM. OTClOlla CneJlyeT, qTO AB AC {::} AC AH .AB.
23. TaK K3K OTpe3KH HP H H Q - BblCOTbl npSlMoyrOJIbHblX TpeyroJlb-
HHKOB ABH H ACH, TO COrJlaCHO nYHKTY 6) npellbl,llYUlew 3allatJH HMe-

nOJl06HbJe TpeyroJlbHHKH
151
eM: AB · AP AH 2 H AG · AQ AH 2 , CM. n.aHHblH HH>Ke PHCYHOK.
CJleJlOBaTeJIbHO, AB · AP AC · AQ. Pa3.ReJIliB o6e tlBCT" 3Toro pa-
AP A
BeHCTBa H3 AB · AC, nOJlYtlHM: AC AB ·
A
Q
B H C
TaK KaK Y TpeyrOJIbHHKOB APQ Ii ABC yroJi A 06lUHH, a CTOpOHbI
AP M AQ nponopUHOH3JIbHbi CTopOHaM AC H AB COOTBerCTBeHHo, TO
TpeyroJlbHHK APQ nOJl06eH TpeyrOJIbHHKY AC B no BTOPOMY npH3Ha-
KY no,no6HH, liTO H TpeOOBanOCb JI.OKa3aTb.
24. PaccMoTpHM JtBa CJlYllaH: I) yroJi npH BepWHHe A OCTpbIH; 2) yroJl
npH BepwHHe A TynoH. B cnyqae I ) OCHOBaHHSI nepneH,U.HKYJlSlpOB AM
H AN nOnaJlalOT H3 npOAOnJKeHHe CTOpOH BC 11 CD 33 TOIJKH B H D,
CM. pHC. I, a B CJlY1.Jae 2) - Ha JIyqH BC 11 DC, CM. pHC. 2.
M B C B
pHC. 1 pRc.2
'N
B cJlYlJae ] ) HMeeM: LMBA LNDA LA, LMAB LNAD
90° LA, LMAN LMAB+LNAD+LA 180 o -2LA+LA
180 0 LA LABG. 3aMeTHM, QTO "3 paBeHCTBa yrJIOB LMBA
M LND A cJlellyeT nono6He npHMoyronbHblx TpeyrOJIbHHKOB ABM H
AM AB AB
ADN. n03TOMY AN AD BC (AD BC KaK n p oT H Bon o-
A
D

152 PemeHHR K  4
.
AM
llo6eH TpeyroJlbHHKY BAC no BTOPOMY npH3HaKY (LM AN LABC,
CTOpOHbl AM H AN nponopUHOHaJlbHbl cTopoHaM AB 11 BC).
CJIYIJaH 2) paccM3TpHBaeTcH notlTH nOJ1HOCTblO aHaJIOrH4HO (npo-
BeD.HTe COOTBeTC1'BYlOlllHe paccY)KJleHHH caMoCTOHTeJIbl-lo). EcnH iKe
LA He HBJ1HeTC5I HH TynblM, HH OCTpbIM, I.e. LA - 90°  TO TOlJKH ]y[
t N COBnaJlalOT C TOlIK3MI1 B H D B 3TOM CJlY4ae TpeyrOJ1bHHKH
A 1\1 N H ABC Boo6111e paBHbI (06n5lCHHTe, n04eMY).
25. PaccMoTpMM l1B3 CJ1YLJ3H: I) yroJ1 LA - OCTpbIH; 2) yroJl LA-
rynoH. B nepBoM cJlytlae LA - 06111HH yroJl npSlMoyroJlbHblX TpeyroJlb-
IHKOB BBIA H CCIA, CM. pHC. I. n03TOMY 3TH TpeyrOJIbHHKH nOLlo6-
AB L AB AB L AC L 'T 6
Hbl, oTKYJta AC] AC 11, 3Ha 1 IHT, AB - AC · 13KI1M 0 pa30M,
ABICI nOJl06eH ABC no BTOPOMY npH3HaKY nOJl06H5I (LA - 06-
lllHH yroJl 3TI1X TpeyrOJ1bHHKOB).
pHC. 1
pHc.2
B
c
B
c
Bo BTOpOM cJIYlJae nOlt06He np5lMoyrOJIbHbiX TpeyroJlbHHKOB B BtA
H CCIA CJleJ!yeT 113 paBeHCTB3 yrJ10B BAB! H CACl KaK sepTHKaJlb-
HbIX. 8 OCTaJlbHOM l{OKa3aTeJlbCTBO aHaJ10rHLJHO CJIY1.JalO I ).
flpU.Ate'laHue. OTMeTHM, LJTO 8 CJ1YlJae OCTporo yrJla A K03Q><pHLlHeHT
nOllo6HSI ABICI H ABC paBeH OTHOllJeHHIO KareTa ABI npSlMo-
yrOJIbHOrO TpeyrOJIbHHKa ABBI K era rHnoTeHY3e AB. 3TO OTHOllle-
HHe H33bIBaerCH KOCHHYCOM oCTporo yr JI3 A.
ECJIH >Ke yroJ1 A 60JIbWe 90°, TO K03Q>Q>HUHeHT nOJ!o6HS1 ABICI
H ABC HBJlSleTCH KOCHHYCOM OCTporo yrJla BlAB (CM. PHCYHOK 2),
,QonOJIHHlOllI.erO yroJl A HO 1800.

nOJl.06Hble rpeyroJlbHHKH
153
26. TaK KaK TpeyrOJIbHI1K ABC paBH06eJlpeHHbIH C OCHOB3HHeM AB,
TO ero MellHaH3 C1\l HBJIS1erCH H BbICOTOH. npSlMoyroJlbHbIe TpeyrOJIb-
HHKH AB H H CBk! nOllo6Hbl, T.K. yrOJ1 np" BepWHHe B y HHX 06111HH,
CM. PHCYHOK.
A M B
AB . B1\l BC .BH. A nOCK O.fl bKY M - cepeHHH3 CTOpOHbl AB, TO
,nOK33aTb.
27. nYCTb ABCD npOH3BOJIbH3H TpaneUHH, P TOllKa nepecelJe-
HHH npOltOJl>KeHHH 60KOBbiX CTOpOH AB H CD 3TOH TpaneLlHH, M H N
- cepen.HHbl OCHOBaHHH BC H AD, Q - TOlIK3 nepeCeqeHHH D-HarOHa-
.neii AC Ii BD. YTBep>Kn.eHHe 3a-na 1 1H 33MeHHM Ha paBHOCHJlbHOe eMY:
TOllKH P H Q Jle>K3T H3 npSlMoH M N.
CHa1la.na ,aOKa>l<eM npHHa.lIJ]eJKHOCTb T04KH P npSlMoH M N. nYCTb
npHM3f1 P AI nepeceKaer OCHOB3HHe AD B TU4Ke N 1 , CM. pl1CYHOK I.
pRC. 1
A
IV.
D
TaK K3K npSlMble BC 11 AD napaJlJleJIbHbl, TO 6BP M nOllo6eH
BM PM C1\1
6APN 1 , a 6CP1\f n O Jl06eH 6DPN 1 - n03 TOMY ANI PN 1 DN 1 .

154
PeweHHH K 4
-
qTO ANI D Nt, T.e. TOllK3 N 1 COBna-naeT C TOQKOM N.
Tenepb ,nOK3>KeM, liTO TOqKa Q Jle>KHT Ha npHMOH M N. llnSl 3Toro
TOqKY nepeCeqeHHSlllHarOHaJlM AC C OTpe3KOM M N 0603HaQHM Qepe3
K 1 , CM. pHc.2a, a TOqKY nepecelfeHMSI !lHarOHaJlH BD c M N o603Ha-
IIHM Qepe3 K 2 , CM. pHC. 26.
pRC. 28)
B M C
pRC. 26)
B M C
AND AND
113 nO.n06HSI 6.CMK 1 H 6.ANI(l HMeeM: MK L CM · a H3 non.o-
NK 1 AN '
H AN D N TO CnpaBeJIJIHBO paBeHCTBO M [( L 1\;1 K 2.. H3 KOTO-
, NK 1 NK 2 '
poro cnenyeT, liTO TOlJKH Kl H K2 COBna.naIOT. n03TOMY ltHarOHaJlM AC
" BD nepeceK3JOT OTpe30K M N B OllHO" 11 TOM >Ke TOllKe K. A no-
CKOJlbKY H llHarOHaJib AC, Ii J{HarOHaJlb BD C0l1ep}l{3T TOqKY 1(, TO
T01JKa K SlBJlSleTCSI TOIJKOH nepecetleHHH AC H B D. Tpe6yeMoe yrBep-
}KJleHlie llOKa3aHO.
28. LJepe3 TOqKY Al npOBeJ1eM npHMYIO, napaJlJ1eJ1bHYIO AB, H TOlJKY
nepeCelJeHHSI 3T0l1 npSlMOH co CTOpOHOH AC 0603HaIJHM Qepe3 B 2 , CM.
u
JlaHHblH "li>Ke PHCYHOK.
AI
A
M
B
TOf1l3 lJeTbipexyroJlbHHK ABAIB2 TpaneUHSI, a T01.JKa C TOlfKa
nepeCeqeHHSI npOJlOJI>KeHHM OOKOBbiX CTOPOH 3TOH TpaneUHH. 113 pe-
u
3ynbTaT3 npe.nhIJlYl..UeH 33Jlall" CJleJ1yeT, liTO T01JK3 nepecelleHMSI OTpe3-

n0,406HbJe TpeyrOJIbHHKH
155
KOB AAI H C M COBnallaeT C TOLJKOH nepeCel.feHHH OTpe3KOB BB 2 Ii
C M, n03TOMY TOLlKa P npHHan..ne>KHT npSIMOH BB 2 "JIM, liTO C3MOe
rJIaBHOe, T01.JKH B, P, B2 J1e)KaT Ha OJlHOH npSlMOH. TaKHM 06pa30M, B 2
3TO TOqK3 nepeCe4eHHH npSlMOH BP CO CTOpOHOH AC, T.e. OHa COB-
nallaeT C TOlJKOH B 1 , llaHHOH B YCJIOBHH. A nOCKOJIbKY AIB2 II AB (no
nOCTpoeHHIO ), TO Tpe6yeMoe yrBep)KeHHe .QOK83aHO.
29. nYCTb AP, BQ - nepneH)tHKYJISIpbl, npOBeJleHHbIe H3 T04eK A Ii
B K npSlMoH C L, CM. JlaHHbIH HH)I{e PHCYHOK.
C
TaK KaK LAC P LBCQ, TO npSlMoyrOJ1bHble TpeyronbHHKH AC P
Ii BCQ nO.Qo6Hbl no nepBOMY npH3H3KY nOJI.06HH. CJIenOBaTeJlbHO,
AC AP
BC BQ.
3aMeTHM, liTO nOCKOJIbKY npHMoyrOJIbHble TpeyrOJIbHHKH AP L Ii
BQL T810Ke nOJl06Hbl (LALP LBLQ KaK BepTHKaJlbHbIe), TO
AP AL AC AP AL
BQ BL · TaKHM o6pa30M, BC BQ BL Tpe6yeMoe pa-
BeHCTBO J10Ka3aHO.
30. nYCTb K Ii N TOIIKH nepeCetleHHH npSlMoH EF c np0J10Jl)KeHH-
HMH CTOpOH BC Ii CD, CM. PHCYHOK Ha CJleJ1}'lOlUeH CTpaHHu e .
TaK KaK AB AE + BE, AD AF + DF H AC AG + CG,
AB BE AD DF AC CG
TO AE 1 + AE ' AF 1 + AF ' AG 1 + AG.  aJIee,

156
PeweHHH K j4
..
K
B
c
C yqeTOM BCeX BblWenpHBelleHHblX paBeHCTB, 113HHOe B YCJ10BHH 3a-
113IJH paBeHCTBO, KOTOpOe H3M Tpe6yeTcSI n.OKa3aTb, MO/KHO npe06pa30-
. BE DF CG
BaTb CJ1e .ayIOUl IfM o 6p a30M . 1 + AE + 1 + AF 1 + AG '
KE+E F +NF CG KN CG
E F ACt EF AG.
3TO cJlellyeT "3 nOllo6HSI
CK KN - T p e6 y e-
AF EF
Moe paBeHCTBO llOK33aHO.
CK
ecJIH cocnaTbcH H8 TOT <fJaKT, liTO OTpe3KH AG M CG COOTBeTCTBeH-
Hhle Jlpyr llPyry 3neMeHTbI nOllo6HbiX TpeyronbHHKOB CK N H AEF.
Ho nOCKOJlbKY 80 MHorHX UJKOJlbHblX y t le6HHK3X flOHSITHe COOTBeTCTBeH-
HbiX 3J1eMeHTOB nOllo(jHbIX rpeyrOJlbHHKOB, OT nHlfHbIX OT CTOpOH H yr-
nOR, He BBO,UJfTCSI, TO 6blno ,naHO 60nee nOJ!po6Hoe 06'bSlCHeHHe 3Toro
paBeHCTBa.
(3TO cJlellyeT H3 no-

TeopeM8 n HcpaJ .OJ)8
157
PemeHHSI K 5 (TeOpeMa nH4JarOpa)
HOMepa 3aOal/" K KOmOpblM npU8eaeHbl peweHUR: 29, 31, 32,
62, 63.
29. nYCTb ABC - rpeyroJlbHHK co CTOpOH3MH 5, 6, 9, npHlJeM AB-
H3H6oJIbUJaH, a AC - HaHMeHbwafi H3 ero CTOpOH, T.e. AB 9,
AC 5, BC 6. VI nycTb H OCHOBaHHe nepneHJlHKYJ1Hpa, npoBe-
JleHHOrO H3 T04KH B K npHMoH AC, a lIJIHHa oTpe3Ka C H paBHa x.
TaK KaK B J11060M TpeyrOJ1bHHKe npoTHB 60JlbWeH CTOpOllbl JlemHT
60JlbWHH yrOJ1, TO LC - H3H6oJlbWHH H3 yr JlOB TpeyroJlbHHKa ABC.
PaCCMOTpHM JIBe B03MO}KHOCTH: 1 ) LC ocrpblH, CM. pHC. I;
2) LC - TynoH, CM. pHC. 2
(CJ1YLJaH LC 90° HeB03MO)l{eH, T.K. MSI TpeyroJlbHHK3 co cTOpOHaMI1
5, 6, 9 He BbillOJlHeH3 TeopeMa nHcfJaropa: 52 + 6 2 =1= 9 2 ).
pHc.1
pRc.2
A
9
B
A
9
B
B 060HX 3THX CJlYlJaHX 113 npHMoyrOJIbHblX TpeyronbHHKOB AB H H
BCH no TeopeMe nHcfJaropa HMeeM: AB 2 AH 2 BH 2 BC 2 CH 2 .
OTJ1HQHe CJIytlaeB I) H 2) ,apyr OT Hpyra COCTOI1T B TOM, liTO B nepBOM
cJlYLJae AH AC - C H 5 x, a 80 BTOpOM AH AC + C H
5 + x. TaKHM o6pa30M, B nepBoM cJlYLJae HMeeM ypaBHeHHe
9 2 -(5 x)2 6 2 -x2,9 2 5 2 +10x 6 2 , lOx 6 2 +5 2 -9 2 ,OTKYJ{a
x -2. 80 BTOpOM CJ1yqae a6C0J110THO aH3J10rHlJHO nOJlYliaeM ypaB-
HeHHe lOx 9 2 - 52 - 6 2 , 113 KOToporo H3XOllHM, 4TO X 2. TaK K31<
3H3 L IeHHe x 2 He y,nOBJleTBopSleT CMblCJlY 33Jt311" (J],flI1H3 OTpe3Ka
He MO>KeT 6bITb orpHuaTeJ1bHa), TO CJIYLJaH I Ha caMOM lleJ1e HeB03MO-
meH. To eCTb Ha caMOM Jle.ne LC - TynoH, npH 3TOM C H x 2,
BH 2 BC 2 - CH2 6 2 2 2 , BH 32 4V2.

158
PemeHlffl K 95
.
n pUMettaHue. ECJlH npHBeeHHble BbIllle paCCY>Kn.eHHH BOCnp0l13BeCTH
lL1IH npHMoyroJlbHoro TpeyroJlbHHK3 ABC, TO B c.TIytlae OCTpOrO yr.na C
Mbl nOJl}'llHM CJlellYlOlllee paBeHCTBO: AB 2 -(AC-CH)2 BC2_CH 2 ,
OTKYHa 2AC · CH AC 2 + BC 2 - AB 2 . A YlJHTbIBa5l, 4TO G1H
BC · cos LC, nonYlJHM TaKoe paBeHCTBO:
2AC · BC · cos LC AC 2 + BC 2 - AB 2 .
CnpaBellJIHBOCTb 3Toro paBeHCTB3 JtJl51 npOH3BOJlbllOrO TpeyrOJlbl1HKa
ABC COCT3BJlSleT )'TBep>K.neHHe «TeOpeMbI KOCI1HYCOB» ( CM. TaKlKe 3a-
llatlY NQ63 Jl3HHOrO naparpacpa).
B TOM cJlyqae, eCJ1H yrOJ1 C TynoH (T.e. B cHTyallHH, H306pa>KeHHOH
Ha pHC. 2), nOJlyqHM paBeHCTBO 2AC · CH (AC2 + BC 2 - AB 2 ).
A YlIHTbIBaH, 4TO CH BC · cos LBCH BC · cos(180° LC), Mbl
no.nYlIHM paBeHCTBO
2AC · BC · cos(180° - LC) -(AC 2 + BC 2 - AB2).
OCTaeTcH 3aMeTHTb, liTO nOCKo.nbKY c()rJ1aCHO onpelteJ1eHHIO KocHHyca
l1J1H yrJIa C, 60JlbWerO 90°, CnpaBellJIHBO TO>KlleCTBO cos(1800-LC)
- cos LC, TO )'TBep>KJleHHe TeopeMbI KOCHHYCOB YCTaHOBJ1eHO HaMH
H B CJlYllae Tynoro yr Jla C.
31. nYCTb ABCD napaJIJleJIOrpaMM co CTopOHaMH AB a, BC b
H OCTpblM yrJloM np" BepllJHHe A (eCJIH yrOJ1 np" BepUJHHe A TynoH,
TO MO)KI-fO npOBeCTH nepeo603Ha 1 leHHe BepWHH). HaM Tpe6yeTcSI !tOKa-
3aTb, lITO Ac 2 + BD 2 2(a 2 + b 2 ). nYCTb BE H CH BbICOTbl na-
paJlJIeJIOrpaMMa, H nycTb DH x, CH y. TaK KaK AB CD (KaK
npoTHBOnOJIO>KHble CTOpOHbl napa.nJ1eJIOrpaMMa) H BE CH (K8K
BbICOTbI napaJlJIeJIOrpaMMa K OJ{HOH H TOH )l{e CTopOHe), TO npSlMoyrOJIb-
HbleTpeyrOJIbHHKH ABE H DCH paBHbl, n03TOMY AE DH X, CM.
u
Jl8HHblH HH>Ke PHCYHOK.
B
b
c
y
A x E
D x H

TeopeM8 nHtjJaropa 159
113 npSIMoyrOJIbHblX TpeyrOJIbHHKOB ACH 11 BDE no TeopeMe
nHcparopa HMeeM:
AC 2 AH 2 +CH 2 (AD +DH)2+ CH 2 (b+x)2+y2,
BD 2 BE 2 + DE 2 BE 2 + (AD - AE)2 y2+ (b x)2.
CJ10>KHB nOLJJ1eHHO Bblpa>KeHHSllJ.JISI AC 2 Ii BD 2 H ytlHTbIBaSl, liTO
x 2 + y2 a 2 (no TeopeMe OHQJaropa H3 CDH), nonyqHM:
AC 2 + BD 2 (b+x)2+ (b x)2 + 2 y 2 2(b 2 +x 2 + y2) 2(1J2+a 2 ),
liTO Ii Tpe6oB3JIOCb IlOKa3aTb.
32. nYCTb ABC - TpeyroJlbHHK co cTopoHaMH BC a, AC b,
AB c, CM me Me)lHaHa 3Toro TpeyrOJIbHHKa. Ha npOllOJl>KeHHH
MellHaHbl CM 3a T04KY M OTJlO>KHM OTpe30K ME, paBHblH CM, CM.
4'
n.aHHbIH HH>Ke PHCYHOK.
A
B
TaK KaK llHarOHaJ1H lleTblpexyrOJIbHHKa ACBE JleJISlTCSI TOIJKOH ne-
peCelJeHHH nOnOJlaM, TO OH HBJlHeTCH napaJlJleJIOrpaMMOM.ll.nHHbl CTO-
pOH 3Toro napaJIneJIOrpaMM3 paBHbl a H b, a JlMarOH3J1H AB " CE
HMelOT JtJIHHY C H 2mc COOTBeTCTBeHHO. COfJI3CHO pe3YJlbTary npeJIbl-
JlYlUeH 33JI.a 1 1H HMe eM: c 2 + ( 2m c)2 2(a 2 + b 2 ), OTKyJ1a
c 2 , me
liTO H Tpe6oB3JIOCb J(OKa3aTb.
c 2
,
62. nYCTb AH - BbICOTa TpeyroJlbHHKa ABC. ECJIH 06a yrJIa B H C
OCTpbIe, TO TOLJK3 H J1e}l{HT H3 OTpe3Ke BC, CM. PlfCYHOK I. ECJIH }l{e
OJlHH 113 yrJloB B H C 1)'nOH (l-IanpHMep, yroJl C), TO TOqKB H JIe>KHT

160
PeweHHR K 5
BHe OTpe3K3 BC, CM.. PHCYHOK 2.
pHc.l
A
pHc.2
A
B
H
c
H
B cJlyqae, H306pa>KeHHOM Ha pHC. I, 113 npHMoyrOJIbHbIX TpeyroJlb-
I-IHKOB ABH H ACH HMeeM: AH AB · sin LB H AH AC · sin LC.
TaKHM o6pa30M, B 3TO M cJlytlae
SIll SIll
B CJ1YLJae, H306pa)KeHHOM Ha pHC. 2, HMeeM: AB · sin LB AH
AC · Sill LAC H. A nOCKOJIbKY B 3TOM CJIYlfae LAC H 180 0 - LC,
TO C · Sill LB b · sin(180° - LC). Ho COrJlaCHO onpeJleJIeHHIO CH-
Hyca Tynoro yrJla CnpaBeDJIHBO paBeHCTBO C · sin LB b · Sill LC <=>
c b
<=> sin LC Sill LB ·
I1T3K, HaMH llOKa3allo , 4TO J1J1 S1 n p OH3B0J1b HOrO TpeyrOJIbHHKa ABC
SIn SIn
113 CTOpOH 3Toro TpeyrOJIbHHKa. B3HB B 310M paBeHCTBe JtPYrylO napy
CTOpOH, HanpHMep, JIe.>KalllHX npOTHB BepwHH A H B, nOJlY4HM, 4TO
a b C a b c
· LA · LB . Jle)!OBaTeJIbHO, · LA · LB · LC '
SIn SIn SIn SIn SIn
liTO It rpe60BaJIOCb ,nOKa3aTb.
63. CM. npHMe4aHHe K perneHHIO 3aJlaqH NQ29 J{aHHoro naparpacfJa.

n.nOlllaJIb
161
PemeHHH K  6 (nJlOllla,l].b)
HOMepa 3aaall, K KomopblM npu8eaeHbl peUlenUfl: 5, 59, 61,
65,72,75,77,79,83,84,86,87,88,89,90.
5. BepwHHbl .naHHoro TpeyrOJIbHHKa 0603HalJHM LJepe3 A, B, C Ii 3a-
MeTHM, lITO rpeyrOJ1bHHK ABC «BnHcaH» B npSlMoyroJlbHHK AK L!vf,
u
CM. JI.3HHbIH HM)I{e PHCYHOK.
L
A
TaK K3K npHMoyrOJ1bHHK AK LM COCT3BJIeH 113 TpeyroJlbHHKOB
ABC, ABK, BCL H ACM, TO era nJlOm3Jlb paBHa cYMMe nJlOm3-
n.eH nepe1.JHc.neHHblx TpeyrOJ1bHHKOB. n03TOMY MCKOM8H nJ10maHb Tpe-
yroJlbHHK3 paSHa SABC SAI(LM - SABK SBCL - SACA1, rIle
1 1
SAKLAf AI(.AJltf 3.4 12, SABK 2 AK.BK 2 .3.1 1,5,
1 1 1 1
S BCL 2 B L · CL 2 . 3 · 2 3, S ACAf 2 AM · CM 2 . 4 · 1 2.
VIT3K, SABC 12 1,5 3 - 2 5,5.
OmBem: 5,5
59. Pell1HM 3aLla4Y LlBYMSI cnoco6aMH.
Cnoco6 1.
nYCTb AM 8x, BN 2y, TOrlta H3 JlaHHblX B YCJIOBHH OTHorne-
HHH CJIen.yeT, liTO CM 7x, CN 3y, CM. JlaHHblH HHiKe PMCYHOK.
C
3y
N
2y
B
A
TaK K3K Y TpeyrOJIbHHKOB AC N H ABC BbICOTa 113 BeplllHHbI A
o6raaH, TO OTHoweHHe nJIOmaJteH 3THX TpeyroJlbHHKOB P3BHO OTHOllle-
6 reOMeTp, 7_8 Kn no..,r K OTJ. TCM. TeCTh.

162 PellleHHH K 96
8 CN  3
HHIO OCHOBaHM" CN  BC, T.e. 8ABC BC 5y 5 · OTCIOJla
3 3
HaXOJlHM, ljTO 8 ACN 5 8 ABC 5 . 100 60.
Y TpeyrOJIbHHKOB C M N H C AN o6111aH BblCOTa 113 BepWHHbl N,
n03TOMY OTHoweHHe HX nJIOlllalleH paBHO OTHoweHHIO C M : CA. 113
S CM 7x 7
paBeHcTBa 8CAN - CA 15x 15 nonYllaeM, liTO 8CAtfN
7 7
15 8ACN 15 · 60 28.
Cnoco6 2.
llJIH HaXO>KJle HHH nJ10111allH TpeyroJlbHH Ka C M N BOCnOJIb3yeMcH
A nOCKOJ1bKY LMCN H LACB - 3TO O)!HH Ii TOT >Ke yro.n, T.e.
· LJ\fCN · LACB SQM.f{ CM · CN 7x · 3y 7
SIn I SIn , TO SABC AC .B C 15x. 5y 25.
7 7
OTclo.na nOJlYlJaeM, 4TO 8e M N 25 8 ABC 25 . 100 28.
Om8em: 28
61. nYCTb PK x, PL y, PM z, PN w, CM. aHHbIH HH-
me PHCYHOK. Tor.na JlJlH nJIOllla,neH npHMoyrOJlbHHKOB, H3 KOTopble npH-
MoyrOJIbHHK ABCD pa36HT npHMbIMH KM H LN, HMeeM paBeHcTBa:
8} x · W, 82 X · Y, 8 3 Y · z, 8 4 Z · w.
B L C
K
8 2 Y S]
x p Z
8 1 W 8
M
A
N
D
nepeMHO)KaH nepBoe H3 3THX paBeHCTB C TpeTbHM, a BTopoe - c
lJeTBepTblM, nOJlyqaeM: 8 1 · 8 3 X · W · Y · Z, 8 2 · 84 X · Y · z · w.
OTCIOJ1a BHJ{HM, LITO 8 1 · 83 82 · 8 4 . TaK KaK no YCJIOBHIO 8 1 36,
S 24 S 48 S 81 · Sa. 36 · 48 72
2 , 3 , TO 4 8 2 24 ·
OmBem: 72

nJlOUlaJ1.b
163
65. nJlOlllallH npHMoyronbHHKOB, Ha I{OTOpbIe npHMoyroJlbHHK ABCD
pa36HT np5lMbIMH K M H LN, 0603HaLJHM lIepe3 8 1 , 82, 83 H 8 4 , CM.
llaHHbIH HH>Ke PHCYHOK. Tor!la 81 · 83 82 · 84 - 3TOT clJaKT llOKa33H B
peweHHH 3a)l3LJH NQ61 n.aHHoro naparpacIJa.
B L C
K
S1. S3
P
SJ S4
M
A
N
D
TaK KaK no YCJIOBHIO 8 1 32, 83 75, a nJlO1ll3)lb npHMoyroJlbHH-
K3 ABCD paBHa 8 1 + 8 2 + 8 3 + 8 4 H paBHa no YCJ10BHIO 207, TO llJIH
HeH3BeCTHbIX 82 If 8 4 liMeeM cHcTeMY ypaBHeHHH:
8 2 + 8 4 + 32 + 75 207 8 2 + 8 4 100
{::>
8 2 · 8 4 32 · 75 8 2 · 8 4 - 2400.
Bblpa3HB 8 4 tlepe3 82 113 nepBoro ypaBHeHHH H nOD.CT3BHB 80 BTopoe
ypaBHeHHe, nonYllHM: 8 2 · (100 - 8 2 ) 2400, 8 - 1008 2 + 2400 0,
OTKY1l3 82 40 HJIH 8 2 60.
OmBem: 40 "JIM 60
72. BOCnOJIb3yeMcH c.ne.nYlOllleH CPOPMYJ10 H D.Jl SI nJIOlUan.H 8 npOM3-
d 1 , d2 - ,ltJIHHbI JlMarOHaJIeH lleTbIpexyrOJIbHHKa, a sinG CHHYC yrJI8
Me>KllY D.HarOHaJIHMH. nYCTb a yroJl Me>KJlY)lHarOH3JIHMH AG H BD
napaJIJIeJ10rpaMM3 AB eD. To rJta, CO rJ1aC HO YKa3aHHoH Bblwe cpOPMY-
8 KL .MN
8ABCD AC .BD ·
-AK - CL
2
7 AC, TO K L
AC-
1
3 BD ,
Jt3HHbIX B YCJ10BHH, nOJlYlJaeM:

164
.
.
1
2
PellleHHH K  6
1
2 BD .
2
7 ' oTKy,na
.
MN BD BM-DN
BD. 1 1 _ 1
6 3
KL MN 4
AC BD 7
4.
ABCD
2 2
SKMLN 7 SABCD 7 . 14
Om8em: 4
A D
OTHOweHHe nJl01l13lteH nOl{06HbIX CPHryp paBHO KB 3 ltpaTY K03<PcfJH-
1 1
LlHeHT3 nO,Qo6HH, n03ToMY SAKN 9 SABD, a SCLA-f 9 SCBD-
CK.na.nbIBaH 3TH llBa paBeHCTBa, nOJIY4aeM:
S
.
(Ha npHBe!leHHOM Bblwe pHCYHKe nJ101l1a.QH TpeyrOJIbHHKOB AK N H
C LM JLTISJ HarJlHJtHOCTH 33WTpHXOBaHbI).
AHaJIOfHlIHO, JLTIH nJl Oll1a J1eH TpeyroJlbHH KO B B K L H DN l'vl nOJIY-
KH BK L H B A C, a TaK>Ke DN M H DAC n O Ho 6Hbi C K03cfJ<pHllH eH -
4
9 SABCD.
3aMeTHM, liTO nJIOlUa.nb LJeTblpexyro.nbHHKa KLMN MO>Ker 6blTb
nOJlYlJeHa BbIlJHTaHHeM 113 n.nOll1alll1 lIeTbIpexyrOJJbHHKa ABCD nJIO-
ll1alleH TpeyrOJIbHMKOB AK N, CLM, BK L, DN M. OTCIO.na, npHllH-
MaH 80 BHHMaHHe nOJIYlleHHbIe paseHCTB8 DJISI CYMM S AK N + SCLM

nJ10m8Jlb
165
H SBKL + SDNM, HaXOJlHM, lITO SKLMN
444
- 9 SARCD 9 SABCD 9 . 117 52.
OmBem: 52
1
SABCD - 9 SABCD
77. nJI01ll311b nJIaHa ropollcKoro CKBepa pasHa CYMMe nJlOLUaJleM npH-
MoyroJlbHHKa ABCD H npHMoyrOJIbHOrO TpeyrOJIbHHKa BCE, CM.ltaH-
u
HblM HM)Ke PMCYHOK.
E
B
A D
5CM 2 ,
TO nJlOllla,nb nJIaHa rOpOl1CKOrO CKBepa paBHa 15 CM 2 .
Bblpa3HM peaJIbHYIO nJIOllla,nb ropo,acKoro CKBepa, paBHYIO no
YCJ10BHIO 60 ra, B CM 2 :
60ra 60.10000M 2 60. 10000. (lOOcM.I00cM) 60. 108 cM 2.
nYCTb k - K03cpcpHLlHeHT, nOKa3bIBaIOLUHH, BO CKOJlbKO pa3 HY>KHO
YBeJJH4HTb nJIaH cKBepa, lITo6bI no.nYllHTb peaJlbHble pa3Mepbl cKBepa.
OTHoweHHe nJlOlllaJleH nOD-06HbIX QJHryp paSHO KSaJ1paTY K03cpCPHUH-
eHT3 nOllo6HSI. n03TOMY OTHorneHHe nJI Olllan.H CKBe p a K nJIO Ula ,aH nJIa-
, 15 CM 2 ,
oTK)'.na k 2 · 10 4 20000. TaKHM o6pa30M, MacwTa6 n-aHHoro
B YCJlOBHH H306pa>KeHHH cKBepa paBeH 1 : 20000.
Om8em: 1 : 20000
79. nYCTb S - nJIOlllaHb Tpey rO JIbHI1Ka ABC, TOrJla nJlOLU3Jlb napaJI-
M K napanJIeJIbH3 AC, a npHM3H N K napaJIJIeJ1bH3 AB, TO Tpe-
yrOJIbHHK BMK nOllo6eH TpeyroJlbHHKY BAG, a CNK n0J106eH
C AB, CM. PHCYHOK H3 CJIellYIOIUeH CTpaHHue.

166
PemeHHH K 6
'II
A
B
K
c
3Ha l lHM llepe3 k. Tor,Ua K03c}J<pHUHeHT n01l06HSI C N K H C AB pa-
BeH C]( BC - BK 1- k.
BC BC
TaK KaK OTHOWeHl1e nJIOl1.lan.eH nOJl06HbIX c}JHryp paBHo KB3J1pary
K03<PclJI1UHel-lTa nOD.06Hst, TO nJIOl1.l3AH TpeyrOJIbHHKOB Bltl K H C N K
OTHOC5ITC5I K nJ101113llH S TpeyronbHHK3 ABC KaK k 2 H (1 - k)2 co-
OTseTCTBel-lHO. 003TOMY SBMK k 2 S, SCNK (1 - k)2S. OTClOlta,
Y 1 IHTbIB351 paBeHcTBo SBlK + SCNK + SAAIKN 8, no.nyqaeM ypaB-
l-IeHHe DJ]H H3XO)K!teHHS1 k:
Pew a H 3 10 KB3JJ.paTHoe ypaBH eH H e , nO Jly qaeM, 4TO k
OmBem: 8 "JIM 24
o.
3
4.
k, TO
4.6
24.
83. OJIOl1.lanH TpeyroJlbHHKOB ABP, BCP, CDP, ADP 0603H3lJHM
lJepe3 8 1 ,82,8 3 , 8 4 , CM. llaHHblH HH>Ke PHCYHOK. no ycnoBHIO, HaM
Tpe6yeTcH .n.OKa3aTb paBeHcTBo 8 1 · 83 8 2 · 84-
8 1
A

nJIOI1l8JJ.b
167
nYCTb LBP A 0:, TOrJla LBPC 180 0 a H no <popMYJle nJIO-
Ula" TpeyrOJIbHI1Ka HMeeM paBeHcTBa:
nepeMHO)KaH nepBoe H3 3TMX paBeHcTB C TpeTbHM, a BTopoe C 4eT-
S2 · 84
TaK KaK sin a sin (180° - a), TO npaBble qaCTH JlBYX npeJlblJtYUlHX
u
paBeHCTB OTJlHtJalOTC5I JlHWb nOpH,llKOM MHO}KHTeJIeH, T.e. paBHbl llpyr
J1PYry, a 3Ha4HT, 8 1 · S3 82 · 8 4 , tJTO Ii Tpe6oBaJ10Cb J10Ka3aTb.
84. I) OYCTb BH H CG BbICOTbl TpaneUHH ABCD, CM. PliCYHOK.
Torn-3,llJIH nJI01UaeH TpeyroJlbHHKOB ABD H ACD HMeeM paBeHcTBa:
1 1
SABD 2 AD · BH, SACD 2 AD · CG- TaK KaK npSlMble BC 11
AD napaJIJleJIbHbl, TO BH CG 11, 3HalJHT, SABD SACD-
B
A H G D
2) 3aMeTHM, liTO TpeyroJlbHHK ABD COCTaBJleH H3 TpeyrOJIbHI1KOB
ABP 11 APD, n03TOMY SABP SABD-SAPD. AHaJlOrHqHO,llJIH Tpe-
yrOJIbHHKa CDP HMeeM: SCDP BACD SAPD- 113 3THX paBeHCTB H
paBeHCTBa SABD SACD cne,nyeT, qTO SABP SCDP, liTO H Tpe6o-
B3J10Cb J10Ka3BTb.
86. 0603HalJI1M 4epe3 S M , S A H S B COOTBeTCTBeHHO nJIOUlaDJt rpe-
yrOJIbHHKOB M CD, AC D H BCD- HaM Tpe6yeTcSllloKa3aTb, liTO
nYCTb AI, Bl, Ml npoeKUHH TOlJeK A, B, M H3 npSlMYIO CD, CM.
u
D-aHHbIH HH)I{e PHCYHOKe

168
PemeHHH K 9 6
M
B
BI C
D AI M 1
UpHMble AA 1 , BBI HMM 1 napaJ1JIeJlbHbl, n03TOMYOTpe30K !\tfM 1
HBJIHeTCH Cpen.HeH J1HHHeH TpaneUl111 A) ABBI. OTclOlia CJ1ellyeT, 4TO
1
!vIM! 2 (AA 1 + BB I ). TaK KaK AA 1 , BB 1 , !vIM! BblCOTbl rpe-
yrOJIbHHKOB ACD, BCD, MC D, TO D.JIH nJ10 ll!a ll e H 3THX rpeyrOJIbHH-
1
SkI 2 A1M l . CD.
CJlO>KHB nOllJleHHO nepBbIe llB3 H3 3THX paBeHcTB, nOJlYlJHM:
1
SA + SB 2 (AA 1 + BB I ) - CD Ml'v[l. CD 2S Af -
8 7 . Cor JI3CHO pe3YJlbTa1)' npeJ!bl.nYll!eH 3aJ(a l lH HMeeM: S lvlCD
1
SICD 2 SABCD CJIellyeT, liTO
SA BCD SACD + SaoD (1).
3aMeTHM, liTO nOCKOJIbKY lleTbIpexyrOJIbHHK ABCD pa3pe3aeTcH
llHarOHaJIblO AC Ha TpeyrOJIbHHKH ACD H ABC, TO
SA BCD SACD + SABC (2)_
113 paBeHCTB (1) H (2) CJle.nyeT, lITO SBCD SABC- TaK KaK y rpe-
yroJlbHHKOB BCD H ABC o6maSI CTOpOH3 BC, a HX n.nOll!a,QH paBHbl,
TO paBHbl BblCOTbJ 3THX TpeyrOJIbHHKOB, npOBeJleHHbIX 113 BeplllHH A H
D K cTopoHe BC.
I1raK, HaMH llOKa3aHO, lITO paCCTOHHHe OT T04eK A H D )to np5lMOH
BC paBHbI, a OTCIO)l3 CJIellyeT, LITO npSIMbIe AD H BC napaJlJIeJlbHbI,
liTO H Tpe6oBaJlOCb ,QOK333Tb.
88. nYCTb M cepe.nMHa 60KOBOH CTOPOHbI AB, a SA, S B, S M nJIO-

nJlOllla,O.b
169
llla,QH TpeyroJlbHHKOB ACD, BCD H MCD COOTBeTCTBeHHO. TaK K3K
paC CT 0 5l HHe d OT T04 K H M 110 npHMoH CD pasHo BblCOTe TpeyroJlb-
TpaneLlHH ABCD pasHa 8 8 A + 8 B TOrJl3 nOJIYIJHM, liTO
828M d .CD.
nYCTb P - T04K3 nepece4eHHH JlHaroHa.neH TpaneuJ-fH. TpeyrOJ1b-
HHK ACD COCT3BJIeH 113 TpeyroJlbHHKOB AP D H PCD, a TpeyrOJIbHHK
BCD COCr3BJIeH H3 TpeyroJlbHHKOB BPC H PCD, CM. PHCYHOK_
B C
A V
CJlellOB3TeJlbHO,8A SAPD+SPCD, S8 SBPC+SPCD 1i,3HalJHT,
8A + SB SAPD + SBPC + 2SPCD- OCTaeTcSI JIHlllb 3aMeTI1Tb, liTO
nOCKOJIbKY SABP SPCD (3TOT Q>aKT YCT8HOBJIeH B 33Jlaqe N284), TO
8 A + SB SAPD + SBPC + SPCD + SABP S - JlOK83aTeJ1bCTBO
Tpe6yeMoro )'TBep)I{,neHH5I 3aBeprneHO.
89. nYCTb AI, B 1 , K 1 - npoeKLlHH TOLJeK A, B, K H3 npSlMYIO CD,
AP Ii AQ nepneHDJiKYJlSlpbl K npHMblM K KI H BBl. HJlHHY oTpe3-
Ka AAI 0603H3lJHM lJepe3 x, a n;II1HY BQ lJepe3 y, CM. AaHHbIH HH)I{e
PHCYHOK.
A
x x x
B. C
D AI K.
k k
CJIeJlyeT, liTO K P 1 + k - y- n03TOMY K Kl x + 1 + k · y. 3aMe-
1 k 1 k
THM, 4TO AA 1 - 1 + k + BB I - 1 + k x. 1 + k + (x + y) · 1 + k
k 1 k
x + y - 1 + k . TaKHM o6pa30M, AAI. 1 + k + BBle l + k KI(l-

170 PeweHHH K  7
k
+ SB. 1 + k SK.
AP D
90. nYCTb BP P, CQ q, rlle p H q HeKOTopble LJHCJIa, H
nYCTb SA, SB, Sp - nJlOLUa-UH TpeyrOJIbHHKOB ACD, BCD, PCD co-
OTBeTCTBeHHO, a Se, SD, SQ nJIOLU311H TpeyrOJIbHMKOB CAB, DAB,
Q AB . Tor n.3 cor JI aCHO pe 3Y JI bT 3ry npe J!b lLlY ll.leH 3alla 1.JH HMe eM :
TaK KaK no YCJlOBHIO Sp SQ, TO nOJlytlaeM CJ1eJlYIOLUee paBeHCTBO:
(*).
A nOCKOJlbKY ABCD rpaneLlHH, TO SA SD H SB Se (CM. pe-
weHHe 3a ,aalfH N284). 11 3 3 TH X ll BYX p aBeHC TB H paBe H C TB 3 (*) CJIeJlyeT,
Sn · (1 + P (1 + q Se. 1 + p) 1 + q)' (8D - Se) · (q - p) O.
PaBeHcTBo (SD - Se) · (q p) 0 MO)l(eT BbInOJIHHTbCH TOJIbKO
B !lBYX CJlYlJ8HX: HJ1H SD Se, HJIH P q. ECJ1M SD Se, TO npHMble
AB Ii CD napanJIeJ]bHbI (CM. 3aHa4Y N285), HO 3TO HeS03MO>KHO, no-
CKOJlbKY 60KOBbIe CTOpOHbl TpaneUHI1 no onpeJleJIeHHIO He napaJlJleJ1b-
Hbl Jlpyr l!PYry. CJleJlOBaTeJIbHO, p q. OTCIO.Qa no TeopeMe, 06paTHOH
reopeMe <l>aJ1eca, Mbl nOJIyqaeM, 1.JTO npHMaH PQ napaJIJIeJ1bHa npH-
MblM BC MAD.
PeweHHSI K  7 (OKPY}I(HOCTb)
HOMepa 3aaatt, K KomopblM npu8eaeHbl petueHUR: 24, 26, 31,
38,42,46,57,61,71,81,89,93,110,120,123,127,133, 134, 145,
151, 153, 157, 165, 166, 167, 171, 173, 175, 178, 179, 180.
24. 3aMeTHM, liTO nOCKOJIbKY LAP D - BHeLUHHH yroJl TpeyronbHHKa
ABP, TO LAPD LBAC + LABD.

OKPYJKHOCTb
TaK K8K BnHC3HHbiM yron paBeH nOJlOBHHe /lY-
rM, Ha KOTOpylO OH OnHpaeTCH, TO LBAC
171
110 0
OTCIOJl3 HMeeM: LAP D 55°+ 77,5° 132,5°.
155 0
n pUMe'laHUe. 113 npHBeneHHblX BblWe paCCY>K.lleHHH CJIellyeT, liTO
113 JJyr, CTSlrHBaeMbIX XOpn.3MH AD H BC.
26. TaK KaK BnHC3HHblH B OKPYiKHOCTb yrOJI pa-
u
BeH nOJIOBHHe rpa,nycHoH Mepbl JlyrH, Ha KOTO-
73,5° 37° 36,5°.
147 0
n pUMettanue. 113 npHBelleHHbIX Bblwe paCCY)KlleHI1H CJleJlyeT, 4TO
113 Jtyr, CTHrHBaeMbIX XOpJI.aMH AD H BC.
31. nYCTb LA 3x, LC 3y, TOr)l3 LB 2x, LD 8y. TaK KaK
4eTblpexyroJlbHHK ABCD BnHC3H B OKPYIKHOCTb, TO LA + LC
LB + LD 180 0 . n03TOMY D)lH HeH3BeTHbiX x, y, BbIpa*eHHblx B
3x+3y 180 x+y 60
rpan-ycax, HMeeM cl1cTeMY: 2x + 8y 180, x + 4y 90. BbIqH-
TBH H3 2-ro ypaBHeHH5I I-oe ypaBHeHHe, HaXOIlHM: 3y 30, y 10,
x 60 y 50. OTCIOJla LVIH yrJloB qeTblpexyrOJIbHHKa ABCD no-
JIyqaeM CJIeJIylOl.I.lHe 3HaQeHHS1: LA 3x 150°, LB 2x 100°,
LC 3y 30°, LD 8y 80°. rpaJJycHaS1 Mepa H3H6oJlbWerO 113
3THX yrJIOB paBHa 150°.
OmBem: 150
38. 3aMeTHM, liTO LBAC LBDC K3K yrJlbl, onHpalOlllHeCH Ha
u
OHHY H TY >Ke Jl.yry, CM. PHCYHOK Ha CJleJJ-YlOllleH CTpaHMu e .

172
PemeHHH K  7
TaK KaK B TpeyrOJlbHHK3X AQC H DQB yroJl 06-
I.UHH, a yrOJ1 A TpeyrOJIbHHK3 AQC paBeH yrJlY D
TpeyrOJ1bHHKa DQB, TO AQC nOJ{o6eH DQB
( no JlBYM paBHblM yr JIaM).
113 nOJl06HH T peyrOJIbHHKOB AQC H DQB c.ne- A
nopLlHH) n0J1Y4aeM: AQ.BQ CQ.DQ, liTO H rpe60BaJIOCb llOK333Tb.
42. JISI pacnOJlO>KeHHH T04KH Q, KOTopaSl HBJ1S1eTCSJ nepeCeqeHHeM
npOOJl>KeHHH CTOpOH AB H CD, B03MO>KHbl ,llBa CJIyqaH:
1) TOlJK3 Q .ne>KMT Ha J1yqax AB Ii DC, CM. pHC. I;
2) TOKa Q Jle>KHT Ha J1Y4ax BA Ii CD, CM. pHC. 2.
C
A
pUC. 2
B
Q
pac. I
D
4ro6bl HartT" JIJlHHY CD, BOCnOJIb3yeMcH CBOHCTBOM ceKYI.UHX:
AQ · BQ CQ · DQ. B CJ1Y4ae, H300paJKeHHOM Ha pHC. I, HMeeM:
BQ AQ AB 56 - 24 32; a B c.nY4ae, H306pa>KeHHOM Ha pHC. 2,
BQ AQ + AB 56 + 24 80. n03TOMY B cJlY4ae I ) nOJIY4aeM, liTO
D Q AQ · BQ 56 · 32 64 C D D Q - C Q 64 - 28 36
CQ 28 ' ·
A B CJI y qae 2 ) HMeeM. D Q AQ · BQ 56 · 80 16 0 n p H 3TOM
· C Q 28 '
CD CQ DQ 28 - 160 -132. Ho nocKonbKY J{JIHHa OTpe3K8 He
MO>KeT 6blTb OTpl1uaTeJJbHoft, TO c.nyqaii 2) H3 caMOM eJJe peaJIH3yeTCH.
OmBem: 36
46. nYCTb AQ x, TOr,ll8 B cJlyqae, H306pa)KeHHOM H3 pHCYHKe I,
BQ x 14, a B CJIyqae, H306pa>KeHHOM H3 pHcYHKe 2, BQ x+ 14.
3aMeTHM, liTO B 060HX cJlY4aHX TpeyrOJIbHHK ACQ nOllo6 eH Tpe -

OKPY)f(HOCTb I 73
AC 15 5
CQ BQ. BD BQ. 18 ' CQ 6 BQ. 003TOMY B nepBoM CJlY-
5 X 22
6 3 ·
pHC. 1
A
Q
pHc.2
B
D Q
BOCnOJIb3yeMcH CHOBa nOJl06HeM TpeyrOJIbHHKOB ACQ M BDQ,
x 5 5 X + 22 a 80 BTO p OM CJI y qae · yp aBHeHHe x 5 5 x- 22
6 6 3' 6 6 3.
25 55 l1x 55
OTCIOD.8 H 8X OAHM, 4TO JI H60 X - 36 x 9 ' 36 9 ' x 20, J1H60
x
u u
OTpHuaTeJJbHa, TO BTOpOH cnyqaH Ha caMOM JteJIe He peaJIH3yeTCSJ, T.e.
x 20.
OmBem: 20
57. nYCTb 0 LleHTp JlaHHOH OKPY>KHOCTH, OM H ON nepneHllH-
KYJlHpbI K xopLlaM AB H CD, CM. PHCYHOK.
C
D
nepneHlI.HKYJlHP, npOBelleHHblH 113 ueHTpa OKPY)KHOCTH K XopJle, e-

I 7 4 PellleHJfH K  7
1 1
J1HT XOpl!Y nOnOJIaM, fl03TOMY HMeeM AM 2 AB , eN 2 CD. TaK
K3K no YCJlOBHIO AP 15, B P 35, C P 21, D P 25, TO
AB AP+BP 50, CD CP+DP 46, AM 25, CN 23 .
no TeopeMe nl1cparOpa 113 CON HMeeM: CO V CN2 + ON2.
JI5I BblllHCJleHHH HCKOMoro palIJ1yca R OKPY)I{HOCTH OCraJIOCb HaHTH
D)lHHY ON.
TaK KaK TPH yrJla 4eTblpexyroJlbHHKa OA.lP N np5lMble, TO 01vfP N
- np5lMoyroJlbHHK 11, 3HalJHT, ON MP. HaHJleM MP:
M P ANI - AP 25 - 1 5 10. OTclO ,na HM eeM : ON 10,
R CO V C N2 + ON2 V2 3 2 + 10 2 V6 29.
OmBem: V 629
61. nYCTb 0 - LleHTp J1aHHOH OKPY)KHOCTH, a
TOLlKI1 M Ii N - CepeJlHHbI OCHOBaHHH AD H
BC COOTBeTCTBeHHO, CM. P H CY HOK. no TeopeMe 0
BHIO, LABD 60° =} LAOD - 120°. TaK K3K
AO DO, TO 0 M - MellHaHa, 6H c c eK TpHca H
11, 3Ha l IHT, cos LCON 0,8. 113 CON HaXOlI.HM:
ON CO · cos LCON 5,6.
TaKHM o6pa30M, BblCOTa MN TpaneUHH pasHa Ok! + ON
OmBem: 9, 1
LBAC
9,1.
71. I) nYCTb P - T04K3 nepeCelJeHHH orpe3KoB
AO H M N, CM. PHCYHOK.
TaK KaK OM H ON pallHycbI K TOIJKaM Ka - A
caHHH, TO LOMA LO N A 90°. TpeyroJlb-
HHKH AOM H AON paBHbl, KaK npHMoyrOJIbHble
TpeyroJlbHHKH c paBHoH fHnoTeHY30H. CJleJ{oBa-
TeJIbHO, AP 6HcceKTpHca yr Jla npH BeplllHHe paBHo6en.peHHoro Tpe-
yrOJIbHHKa ", 3HalJHT, AP MellHaHa H BblCOTa TpeyrOJIbHHK3 AM N.

OKPY»<HOCTb
175
2)MP
AO - OP 40 - x. KaK H3BeCTHO, KBaJIpaT BblCOTbl npHMoyrOJIb-
Horo rpeyrOJIbHHKa paBeH npoH3BelleHHIO oTpe3KoB, H8 KOTopble BbICO-
T3 ,QeJIHT rHnoTeHY3Y. n03TOMY Mp2 AP · OP, 19,2 2 (40 x)x,
x2-40x+368,64 0 <=> x 14,4 HJ1H x 25,6. TaK K3K no YCJlOBHIO
OM > AM, TO OP > AP. npM x 14,4 YCJIOBHe OP > AP He Bbl-
nOJlHeHO (OP x 14,4, AP 40 - x 25,6). n03TOMY x 25,6.
no TeopeMe nHcparopa H3 TpeyroJlbHHKa OMP HMeeM:
01\;[2 Op 2 + Mp2 25,6 2 + 19,2 2 1024, OM 32.
OmBem: 32
81. nYCTb ABC .naHHbIH paBHo6e-npeHHblH TpeyroJlbHHK C OCHOBa-
HHeM AB (DJIHHY KOToporo HaM Tpe6yeTcH H3HTH), TOlJK3 0 ueHTp
era onHcaHHOH OKPY)KHOCTH, a M CepeltHHa OCHOBaHHH AB. no-
CKOJlbKY Me;!H3Ha CM K OCHOBaHHIO AB paBH06el1peHHOro TpeyrOJ1b-
HHKa ABC SlBJlHeTCSI cepellHHHblM nepneHD.HKYJIHpOM K AB, TO TOLJKa
o J1e>KHT Ha npHMOH CM. nYCTb OM x. PaccMoTpMM ,QBa B03MO>K-
HbIX CJ1Y4aH: 1) LC OCTpbIH, CM. pHC. I; 2) LC - TynoH, CM. pHC. 2.
pRC.!
pHc.2
A
A
M
B
B cJlYlJae I) 113 TpeyroJlbHHKOB ACM H AOM no TeopeMe nH<fJa-
ropa HMeeM: AM 2 AC2 - CM 2 A02 - OM 2 , 80 2 - (50 + x)2
50 2 x 2 , lOOx 80 2 - 2 · 50 2 , X 8 2 - 2 · 52 14. A B CJIY-
LJae 2) a6COJIIOTHO aHaJIOrHlJHO nOJlyqaeM: 80 2 - (50 x)2 50 2 - x 2 ,
x -14. TaK KaK JlJlHH3 oTpe3Ka OM He Mo*eT 6blTb OTpHuaTeJlb-
HOM, TO CJ1YllaH 2) Ha caMOM ,lleJIe y*e HeB03MO>KeH. TaKHM o6pa30M,
x 14, OTKy.n3 AM2 A0 2 Olv1 2 50 2 - 14 2 2 2 · (25 2 7 2 )
2 2 · 24 2 48 2 , AM 48, AB 2AM 96.
OmBem: 96

176
PellleHHH K  7
89. OYCTb 2x ltJ]HHa MeHbwero H3 OTpe3KOB, Ha KOTopble rHnOTeHY3a
1{eJlHTCSI T01.JKOH KacaHHH, TOf,na 3x - .unHHa 60JIbWerO 113 3THX OTpe3-
KOB. OOCKOJ1bKY OTpe3KH KaCaTeJIbHblX K OKPY.>KHOCTH, npOBelleHHble 113
O.nHOH TOLlKH, paBHbI, TO CTOpOHbI TpeyrOJ1bHHKa 4 4
HeJlSlTCH T04KaMH K8CaHHH TaK, K3K nOKa3aHO H3 2x
pHCYHKe. OTCIOJta, npHMeH5IH TeOpeMY nH<jJarO- 3x
pa, nOJ1YlfaeM ypaBHeHHe:
(5X)2 (3x + 4)2 + (2x + 4)2.
-
x
pasHa 5x 20.
Om8em: 20
4. KopeHb
4 rHnoTeHY33
93. nYCTb K, L, ]\iI, N - TOllKH KaCaHH5I BnHcaHHOH
co cTopoHaMH AC, BC, AB H npSlMoH PQ co-
OTBeTCTBeHHO, CM. PHCYHOK. TaK K3K nepHMeTp Q
TpeyroJlbHHKa CPQ paBel 20, TO H3 paBeHCTB
PK PN, QL QN nOJIY1IaeM: C/( + CL
CP+PN+CQ+QN CP+CQ+PQ 20.
113 paBeHCTB AK A.NI H BL B1vI HMeeM: A M
AK + BL AB 15. OTCIOl!a HaXO,QHM, liTO nepHMeTp 6ABC
paBeH AC + BC + AB (CI( + CL) + (AK + BL) + AB
20 + 15 + 15 50.
11 3 n o,no6 H SI TpeyrOJIbHHKOB PQC H ABC CJIeltyeT, llTO OTHo weHHe
OmBem: 6
11 o. nYCTb TOQKa 0 ueHTp OKPY}I(HOCTH, BnHcaHHOH B MHoroyroJlb-
HHK, AB Ii BC cocellHHe CTOpOHbI MHoroyrOJIbHHKa, K KOTOpbIM npo-
BelleHbI paJtHycbI BnHcaHHOH OKPY>KHOCTH, K, L TOqKH KacaHHH BnH-
caHHOH OKPY>KHOCTH C 3THMH CTopOHaMH, CM. PI1CYHOK cnpaBa. TaK KaK

OKPY)I(HOCTb
177
JIO CTOpOH MHoroyrOJ1bHHKa. 113 ypaBHeHHSI
N 60.
OmBem: 60
MHoroyrOJIbHHK npaBHJIbHbIH, TO BO M CO pa-
JIHYCbI ero OnMCaHHOH OKPYIKHOCTH, T.e. BO CO.
003TOMY TpeyrOJIbHHKH BKO H CLO paBHbl, H,
3HalJHT, LBOK LCOL. OTCIO!la c.nellYeT, 4TO
LBOC LLO[(. no YCJIOBI1IO LLOK 6° , TaKHM o 6pa30M,
L C
QHC-
120. TaK K3K B [leTblpexyrOJIbHHK BnHcaHa OKPY>KHOCTb, TO CYMMbl
.llJIHH era npOTHBOnOJlOiKHblX CTOpOH paBHbI. n03TOMY n..nHHa '-IeTBep-
TOH CTOpOHbl 6YJ!eT HaHMeHbwet't B TOM CJIYIJae, eCJIH npOTHBOJIe>K3Ill3H
eH CTopOHa HMeeT H3H60JlbWYlO JlJIHHY.
nYCTb x llJIHH3 lleTBepTOH CTOpOHbl. 113 CKa3aHHoro BbIwe cJle-
JlyeT, liTO x HMeeT H3HMeHbwee 3HaqeHHe B TOM CJ1Y4ae, eCJIH YllOBJIe-
TBopHer ypaBHeHHIO: x + 101 90 + 99, OTKYJl3 x 189 101 88.
OmBem: 88
123. CYIllecTBYIOT ,l!Be OKPY>KHOCTH c ueH-
TpOM B TOllKe B, KacalOlllHeCH llaHHoH B YCJIO-
BHH OKPY)KHOCTH, o6e OHM H306pa>KeHbl H3
..
npHBeJleHHOM cnpaB3 pHCYHKe nYHKTHpOM.
OYCTb C T04K3 KacaHH51 H3H6oJIbWeH
v u
H3 3THX OKPY>KHOCTelf C }!3HHOH B YCJlOBHH
OKPY*HOCTblO.
.nH paJlHyca BC 3TOH OKPY>KHOCTH HMeeM paBeHCTBO:
BC AB+AC 10+7 17.
C 7
A
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OmBem: 17
127. nYCTb TOqK3 P ueHTp MeHbllleH OKPY>KHOCTH, a TOllKa 0
ueHTp 60JlbWeH. TaK K3K MeHb1ll3S1 OKPY>KHOCTb K3caeTCSI 60JlbWeH B
T04Ke C, TO TOlJKH C, P H 0 JIe>KaT Ha OJtHOH npHMoH, CM. PHCYHOK Ha
u
CJIeD.YlOlU eH CTpaHHue.

178
PeweHHH K  7
"
nOCKOJlbKY OA OC Ii PB PC (KaK
pa!lHycbl OKPY)KHOCTeH), TO LO AC LOC A
H LPBC LPCB, a nOCKOJIbKY LPCB H
LOC A 3TO OD.HH H TOT )Ke yrOJI, TO LOAC
LPBG. OTCIOD.a CJIeJ1yeT, liTO TpeyronbHHKH
OAG H PBG nOJl06Hbi (no LlBYM paBHblM yr-
JIaM ).
TaKHM 06 p a30M , AG OA 12 A c
BC P B 5'
16,8, AB AG BC 16,8 - 7 = 9,8.
OmBem: 9,8
133. I) no reopeMe OHcparopa, AC 2 AB 2 - BC 2 225, AC 15.
COrJ1aCHO H 3 BecTHoH cpopMYJIe, paJtHYc OKP Y )K HOCTH, BnHc3HHo H B npSl-
u
ueHTp BnHcaHHOH OKPY)K-
HOCT" ABC, TOqKa 0 1 LleHTp OKPY)KHOCTH, KacalOllleHC5I npHMbIX
AB, AC H K8C31OUleHCH BnMcaHHOH OKPY*HOCTH 6.ABC, a x - ,UJIHHa
u
paD-Hyca 3TOH OKPY)KHOCTH.
Ka>KJlaSl H3 TOlJeK 0 H 0 1 paBHoyaJIeHa OT npHMbIX AB, AC, H
n03TOMY JIe}l{HT Ha 6HcceKTpHce yrJIa CAB. OTMeTHM cpa3Y, 4TO 803-
MO}l{HbI JlB3 CJ1Y43S1 pacnOJIO>KeHHH TOlJKH 0 1 - Ha orpe3Ke AO 1-1 BHe
3Toro OTpe3Ka, CM. pHc.la) Ii 16).
1 a) C 1 6)
c
12 BC
5
12 .7
5
A
B
A
B
OlJeBHAHO, lJTO B nepBoM cJlYlJae x < r, a 80 BTOpOM cJlYLJae x > T.
2) HaH!leM HCKOMYIO BeJIHlJHHY x J]JISI Ka>K!lOrO H3 CJlYlJaeB. npH
3TOM onHpaTbCSI 6yn.eM H3 PHCYHOK ,ll.nH CJIY1.JBSI x < T, a TeKCTOM B
CKo6K3X 6YJleM YKa3bIBaTb Te H3MeHeHH5I, KOTopbIe 6YllYT npOHCXO,QMTb
B CJIYl.Jae x > r.
nYCTb P, M TOIIKH KacaHHH npHMoH AB C BnHcaHHOH OKPY>K-

OKPYHOCTb 179
HOCTblO 6ABC H TOH OKPY>KHOCTblO, paJlHYC KOTOpOH Mbl HllleM, K
TOIIK3 KaC3HHH BnHcaHHOH OKPY)KHOCTH 6ABC " K3TeT3 AC, H
npoeKUHH TOllKH 0 1 Ha npSlMYIO OP, CM. pHc.2.
puc.2
AMP B
3aMeTHM, liTO nOCKOJIbKY OKPY)KHOCTH, H306pa)KeHHbIe Ha pHCYHKe, Ka-
calorCH Jlpyr llpyra, TO 00 1 r + X X + 3. CocTaBHM ypaBHeHHe AJIH
HaXO)KJleHHS1 HeH3BeCTHoro x, onHpaHcb Ha nOJl06He TpeyroJlbHHKOB
KH 0 H, COOTBeTCTBeHHO, OH PH - OP x 3). HatiJteM AP
H 3aTeM BbI4HCJIHM O A: AP AK AC CK AC - r 12,
OA V Ap2 + Op2 3 17. BblnOJIHHH cooTBeTcTBYlOlllHe nOJlCT3-
nOJlyqaeM ypaBI1eHHe x + 3 17 · x - 3 17). OTCIOJla H3XOj{HM:
x 3 . 17 1 3 . (17 - 1)2 3. 9 - 17 .1leH-
17+1 (17+1).( 17 1) 8
.
134. nYCTb 0 ueHTp OKPY)KHOCTH, BnHcaHHOM B TpaneUHIO ABCD,
a K - TOllKa KacaHHH 3TOH OKPY)KHOCTH C OCHOBaHHeM AD, CM. PHCY-
A.iVi 10 2 8, a nocKonbKY AK AM (KaK oTpe30K KaC3TeJlb-
HbIX, npOBeJleHHbIX H3 OJ1HOH TOlfKH), TO AK 8, DK AD- AK 16.

180
PellleHHH K  7
..
c
8 K 16 D
pHC.! p uc.2
BaTeJlbHO, ABO - npHMoyroJlbHbIH TpeyrOJIbHI1K C rHnoTeHY3011 AB.
TaK KaK OM 1.. AB (KaK pa.nHYc, npOBeJ1eHHblH B TOqKY KaCaHHH), TO
oTpe30K OM HBJlHeTCH BblCOTOH 6ABO, H no CBOHCTB Y BbI COTbl npH-
MoyrOJIbHOrO TpeyrOJ1bHHKa HMeeM: OM V A M · BM v S · 2 4.
3HaH J]J1HHY oTpe3Ka DK 11 paJlHyca OK, HaHlleM MCKOMble paHH-
YCbl r H R OKpY>KHOCreH, KaC310111HXCSI CTOpOH yrJl3 CDK Ii KacalO-
LUHXCH ,aaHHoH OKPY>KHOCTH c ueHTpoM B TOtIKe 0, CM. PHCYHOK 2. IlJIH
..
3Toro J:lOCT3rOlJHO nOBTopHTb BbIKJJa!lKH, npoBeAeHHbIe npH BblqHCJle-
HliH HCKOMoro palLHyca B npeJlblJlYlUeH 3a.nalJe. npoBe.QH cooTBercTBY-
9 - 17 R 9 + 17
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145. nYCTb paJlHYc MeHbweH OKPY)I{HOCTH paBeH 20x, Torlla pallH-
yc 60JIbUleH OKPY>KHOCTH paBeli 21x, a nJIOlllaDJi MeHbweH 11 60JIbUJeH
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HorD 3THMH OKPY>KHOCTHMM, pasHa 4411rx 2 - 4001fx 2 417rx 2 , 1JTO
no YCJIOBHIO COCTaBJIHeT 3697r. OTclOlla HaXOJ!HM, liTO x 2 9, x 3,
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Om8em: 63
t 51. 3aMeTHM, qTO Jl)lSI nJlOlUaJlH S H llJIHHbl !lyrH L ceKTopa Kpyra
CnpaBeJ]J1HBbl <P0PMYJIbl: S  . R2, L Q · R, rJle R paJlHYc
Kpyra, a pallHaHH3H Mepa yrJIa 3Toro ceKTopa. 113 3THX llBYX CPOPMYJ1

OKPY)I(HOCTb
181
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7,5, S 150. Om8em: 150
153. OYCTb P nepMMeTp M H o ro yrOJlbHHKa, S ero nJlOmB)1b. To-
royrOJIbHHKa BCer,lla 60JIbllle, qeM nJlOma,ab BnHC3HHoro B Hero Kpyra
(eCJIH Boo6111e B Hero MO>KHO BnHcaTb Kpyr). 3rOT cpaKT Har JIHJ{HO npo-
HJlJlIOCrp"pOBaH H3 JlaHHOM HH>Ke pHCYHKe, rj!e «H30bITOK» nJlOllla.aH
MHoroyrOJIbHHKa no cpaBHeHMIO C n.,10111a.QblO BnHcaHHoro B Hero Kpyra
v
BbIlleJIeH WTPHXOBKOH.
 
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TaKHM 06pa30M, S > 1rr 2 H, 3H3 t IHT,
P > 27fT, lfTO Ii Tpe6oBaJlOCb n.OK33aTb.
157. TaK KaK LleHTp BnHcaHHOH OKPY>KHOCTH TpyrOJIbHHKa HBJIHeT-
Cst TOqKOH nepeCelJeHHSI era 6HcceKTpHC, TO AO Ii BO 6HcceKTpMcbl
B HyrpeHH H x yrJIOB np" Be pw HHax A H B. PaBeHcTBo LAOB
Ka3aHO B peweHHH 3311aqM NQ52 H3  3.
165. npOBeJleM Qepe3 TOlJKY C npHMYIO l, KacalOIUyIOCSI aD-Hoft 113
Ol<pY)KHOCTeii. TaK K3K OKPY>KHOCTH K3C31OTCH pyr Jlpyra B TOqKe C,
TO 3Ta npHMaH I 6y,neT TalOKe KacaTbCSI Ii JlpyroH OKPY>KHOCTH ( 06DHC-
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npSlMoH AB, CM. PHCYHOK.

182
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Hoff TOlJKM K OKPY>KHOCTH. 11, aHa.nOrHlJHO, B N C N. n03TOMY AN
CN B N, T.e. CN - MeDJi3Ha TpeyrOJIbHHKa ABC H, KpoMe TO-
cJIYlJae LAC B 90° .
TO LBAG LADG, CM. 113HHbIH HH>Ke PHCYHOK.
D
A B
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BaJlOCb ,nOKa3aTb (paBeHCTBO LACB 90° llOK33aHO B npeJI.blnymeM
pellleHHH ).
167. n poseJ{eM npHMYIO AP, K3calOrnylOcSI OJlHOH 113 OKpY)I{HOCTeii,
HanpHMep, MeHbweH, CM. l!aHHbIH HH/Ke PHCYHOK. TaK KaK OKPY>KHOCTH
K3C31OTcSlllpyr n.pyra B TOlJKe A, TO npHMaH AP 6y,aeT T310Ke KaCaTbCs:I
H 60JIbllleH OKPY}KHOCTH (06'bSlCHHTe caMOCTOHTeJIbHO, nOlJeMY).
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OKPYHOC 183
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LP AB. A nOCKOJIbKY LP AB H LP AD 3TO O)lHH Ii TOT >Ke yroJl,
TO H3 JtOK333HHbIX BblWe paBeHCTB LAED LP AD, LACB LP AB
CJle,ayer, liTO LAED LACB. TaKHM 06pa30M, paBHbl COOTBeTCTBeH-
HbIe yrJIbl npH nepeCeqeHHH npHMbIX BO H DE npHMoH AC H, 3H3lJHT,
3TM npHMble napaJ1J1eJ1bHbI, liTO Ii Tpe6oBaJIOCb .LlOKa33Tb.
171. 4T06bl HOKa33Tb paBeHcTBo PAl BA 1 , ,nOCTaT04HO .nOKa3aTb,
liTO B TpeyroJlbHHKe BP Al paBHbl yrJlbl npH BepwHH3x B H P. nYCTb
Q, {3, , - BeJ1H4HHbl yrJIOB npH BeplllHHax A, B, C TpeyroJlbHHKa ABC.
TaK KaK eHTp snHcaHHOH OKPYHOCTM - 3TO T04K3 nepece4eHHH
6HcceKTpHC, TO LBAP LCAP a/2, LABP LCBP /3/2,
CM. PHCYHOK. YrOJI BP Al HBJIHeTCSI BHeWHI1M yrJloM TpeyrOJIbHHKa
ABP, n03ToMY LBP Al LBAP + LABP 0!/2 + {3/2.
A
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3aMeTHM, liTO nOCKOJIbKY LA1BC LCAA 1 0:/2 (KaK yrJlbl,
onHpalOUlHecfI H3 OJ{HY JlYry), TO LAIBP LA1BC + LCBP
a/2 + /3/2. BcnOMHHaH, liTO LBP Al 0./2 + /3/2, nOJlyqaeM, qTO
LAIBP LBP AI, TeM caMblM .llOKa38TeJlbCTBO Tpe6yeMoro }'TBep-
}KJleHHSI 3aBeprneHo.
173. npoBe.neM OKPY>KHOCTb, co,nep>Kal.llYlO TOqKH A, B H C (3TO Bce-
rJla B03MO>KHO, T.K. BOKpyr JlI06oro TpeyrOJIbHHKa MO>KHO onHcaTb OK-
PY>KHOCTb ).llonYCTHM, liTO LACB LADB, HO TOqKa D He Jle>KHT Ha
3TOH OKPY>KHOCTH. Torn.a TOlfKa D Jle>KHT JI"60 BHe 3TOH OKPY>KHOCTH,
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184 PeweHHH K 9 7
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pHC.!
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CTOPOHY OT npHMOM AB, TOJIbKO B 310M CJ1Y4ae AB - 310 Ta >Ke CaMa51
Jlyra, liTO H B npe,llbIJlYWJiX paBeliCTBax llJlH LADB). TaKHM 06pa30M, B
LACB BbInOJ1H5ITbCfI He MO>KeT. nOJlyqeHHOe npOTHBope4He nOKa-
3bIBaeT, liTO Hallie npennOJIO>KeHHe 6blJIO HeBepHO, T.e. TOlJKa D J1e)KHT
H3 onHcaHHoH OKPY>KHOCTH TpeyroJlbHHKa ABC, liTO H Tpe6oBaJIOCb
JtOKa3aTb.
175. nYCTb 0 - T04Ka nepeCe4eHH5I 6HcceKTpHc BHYTpeHHHX yrJIOB
npH BeplllHH3x A Ii B BblnYKJIOrO QeTbIpexyrOJIbHI1K3 ABCD, CM. pHC. I.
F C
A
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Tor.na T01JKa 0 paBHoYJ1aJleHa OT CTOpOH AB H AD (T.K. JIe)l{HT
Ha 6HcceKTpHce yrJIa A) H paBHOYJlaJleHa OT CTOPOH AB M BC (r.K.
Jle>KHT H3 6HcceKTpMce yrJla B). n03TOMY TOlJKa 0 paBHoYJlaneHa OT

OKPY)/(HOCTb
185
Tpex CTOpOH AB, AD, BC. nYCTb h paccToflHHe OT T04KH 0 llO
3THX CTOpOH. n pO Be.neM OKPY>KHOCTb pallHyca h c u.eHTpOM 8 TOqKe 0,
3Ta OKPY>KHOCTb 6YJleT KaCaTbCH TpeX npHMblX AB, AD M BC.
npeJ!nOJlO>KMM, LITO AB+CD BC+AD, HO npHMaSi CD He K8-
caeTCH nOCTpoeHHOH H3MH OKPY>KHOCTH. TOfJla npHM3S1 CD J1H60 nepe-
ceKaer nocTpoeHHYIO OKPY>KHOCTb, JlH60 He HMeeT C HeM 06111HX TOlJeK.
B OUOHX YK33aHHbix CJIY4aHx npOBelleM KaCaTeJIbHYIO K nOCTpoeH-
HOH OKPY)KHOCTH, napaJlJ1eJIbHYIO npS1MoH CD, Ii T04KH nepeCelJeHHH
3TOH KaCaTeJlbHOH c npSlMbIMH AD Ii BC 0603HalJHM lJepe3 E H F CQ-
OTBeTCTBeHHO, CM. PHCYHKH 2 H 3.
nOCTpoeHHaH HaMH OKPY>KHOCTb KacaeTCH Bcex CTOpOH lJeTbIpex-
yronbHHK3 ABFE (T.e. BnHcaHa B Hero), n03T0M)' no CBOHCTBY 4eTbl-
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pexyrOJlbHHKa, B KOTOpblH MO>KHO BnHcaTb OKPY)KHOCTb, MMeeM:
AB + EF BF + AE. BblllHT3S1 113 3Toro paBeHCTsa JlaHHOe no YCJlO-
BHIO paBeHCTBO AB + CD BC + AD, B CJlY4ae, H306pa)l{eHHOM H3
pHc.2, HMeeM: EF - CD (BF BC) + (AE AD) CF + DE,
OTKY.lla EF CD + CF + DE. Ho 3TO HeB03MO>KHO, T.K. J])1HHa J10-
M3HOH FC D E 60J1bWe JlJIHHbl oTpe3Ka EF (06DHHHTe 3TO nOJlpo6-
Hee, OCHOBbIB3S1Cb H3 HepaBeHcTBe TpeyrOJ1bHHKa). n03TOMY CJIY1IaH,
H306pa>KeHHb1H 113 pHcYHKe 2, Ha C3MOM JleJIe HeB03MO>KeH. nOK3>KH-
Te caMOCTOHTeJIbHO, liTO CJlY I 13H, H306pa)KeHHbIH Ha pHCYHKe 3, T310Ke
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paBeHCTBO AB + EF BF + AE).
TaK KaK 06a CJIY4a5l, npH KOTOpbIX npHMafi CD He KacaerCH no-
u
CTpoeHHOH HaMH OKPY>KIIOCTH, HeB03MO>KHbl, TO nOCTpoeHH3S1 OKPY)K-
HOCTb KacaeTCH npHMoH CD, T.e. BnHC3Ha B lIeTblpexyrOJ1bHHK ABCD,
QTO H Tpe6oBaJ10Cb ,QOKa3aTb.
178. CHalJaJIa paCCMOTpHM TOT CJIyqati, KOr)la Bee yrJ1bI TpeyrOJIbHM-
Ka ABC OCTpble. nYCTb TOlJKa P cepellHHa oTpe3Ka AH, a TOqKa
M cepe.n HH 3 CTOpOHbl BG. llOCT3TOllHO JlOKa33Tb, liTO TOqKH P "
M JIe>K3T Ha onHC3HHOH OKPY>KHOCTH TpeyrOJIbHHKa A1BICl.llJlH TO-
qeK, SlBJIHIOlUHXC5I Cepe!lHHaMH OTpe3KOB BH, CH M Cepe!lHHaMM CTO-
POH AC Ii AB .nOKa3aTeJlbCTBO 6y.neT aHaJIOrHQHbIM.

Pel1leHHH K  7
nYCTb a, /3, 'Y rpaJlycHbIe Mepbl yrJloB A, B Ii C COOTBeTCTBeH-
HO. TaK KaK LACIH LABIH 90°, TO OTpe30K AH HBJIHeTCfI 11"a-
MeTpOM OKPY>KHOCTH, onHcaHHOH BOKpyr QeTbIpexyrOJ1bHHKa ACt H Bl.
n03TOMY cepe.nHHa OTpe3K3 AH (T.e. TOIlKa P) HBnHeTCH LleHTpOM onH-
caHHOH OKPY)KHOCTH ABI G 1 Ii, 3HatJMT, LC1P B 1 2LC 1 AB 1 = 2a,
CM. PHCYHOK 1.
186
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pHC. 1
B AI C
nOCKOJlbKY qeTbIpexyroJlbHHK BC I HAl BnHcaH B OKPYiKHOCTb (aHa-
JlOrHIJHO 4eTbipexyronbHHKY AG1HB 1 ), TO LHAIG] LHBG 1 , KaK
yr JIbl, onHpalOLUHeCH Ha OJ!HY H TY )Ke yry. AHaJlOrHlJHO HMeeM:
LH AIBl LHCB 1 , CM. pHC. 1. 3aMeTHM, qTO LBIBA LC}CA
90° - Q. CneJlOB8TeJIbHO, LH Al C t LH AIBI 90° - Q H, 3Ha-
lJHT, LBIAI C l 180 0 - 20.
B CHJlY paBeHcTB LC1P B 1 2a 11 LBIAI C t 180 0 - 2a HMeeM:
LCt P B 1 + LBIAI C l 180 0 . OTCIOJla CJle,nyeT, liTO qeTblpexyrOJIbHHK
AIBIPG I BnHcaH B OKPY>KHOCTb HJIM, HHalle rOBopH, TOllKa P JIe>KHT
H3 onHcaHHOH OKPY)KHOCTH AIBI C l .
Tenepb ,llOKa.>KeM, liTO TOlJKH M, AI, B 1 , 0 1 J1e>KaT H3 OllHOH OKPY)K-
HOCTH.1!JIH 3Toro D.OCTaTOlJHO nOKa3aTb, liTO LB I GIM + LBIA1M
180 0 . TaK KaK M - cepeHHa rHnOTeHY3bI npSlMoyrOJIbHOrO TpeyrOJIb-
HHKa BGCt, TO GIM CM Ii, 3HaQHT, LMC1C LMCC 1
90° - {3, CM. PHCYHOK 2.
pHC. 2
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B M Al C
T3K KaK qeTblpexyrOJIbHHK B1ACIH BnHcaH B OKPY>KHOCTb, TO

OKPY)/(HOCTb
187
LB1CIH LBlAH LCAA 1 90°-,_ 113paBeHcTBa LB}G1H
90° , H paBeHcTBa LM GIG 90° - (3 nOJlYI.JaeM, liTO LB1C1M
180° - (3 ,. 3aMeTHM, liTO LJeTbIpexyrOJIbHHK ABAIBl BnHCaH
B OKPY>KHOCTb (TOIlKH Al H B 1 HBJIHIOTCH BeplllHHaMH npHMbIx yrJIOB,
OnHpalOLl.lHXCff H3 oTpe30K AB, Ii n03ToMY J1e)K3T Ha OKPY>KHOCTH, no-
cTpoeHHoH Ha oTpe3Ke AB KaK H3 JlHaMeTpe). TaK KaK qeTblpexyroJlb-
HHK ABAIBI BnHcaH B OKPY>KHOCTb, TO CYMMa era npOTHBOnOJlO>KHbIX
yrJ10B paSHa 180 0 , n03ToMY LBIAIB 180 0 - Q.
113 paBeHCTB LB I CIM 180 0 {3 - , " LBIA1M 180 0 a
CJIe-nyeT, lITO LB1C1M + LBIA1M 360 0 Q - (3 - , 180 0 .
TaKHM o6pa30M, Tpe6yeMoe paBeHcTBo llOKa3aHO, 11 TOqKH M, AI, B 1 ,
C 1 Jle>KaT Ha OJI.HOH OKPY>KHOCTH.
llJl51 3aBeprneHHff JlOKa3aTeJIbCTB8 YTBep}l{D.eHHH nYHKT3 a) oCTaeT-
u
C5I J1HWb paCCMOTpeTb TOT CJIYlJaH, Kor,na OJ!HH 113 yrJIOB TpeyrOJIbHHKa
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3aMeTHM, liTO B 3TOM CJIYlIae TpeyrOJIbHHK BC H HBJIHerCH OCTpO-
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Ka BC H, a TOIIKH Ai, B 1 , G 1 OCHOBaHH5I BbICOT 3Toro TpeyrOJIbHH-
Ka. B cJIyqae oCTpoyrOJIbHOrO TpeyrOJIbHHK3 YTBep)KAeHHe nYHKTa a)
Y)I{e J!OKa3aHO. npHMeHHB 3TO YTBep>KJleHHe K TpeyrOJIbHHKY BCH,
nOJIYLJHM, liTO cepellHHbI OTpe3KOB BA, CA, H A H CepeJlMHbl oTpe3-
KOB BC, BH, CH npHHal1J1e>K3T onHcaHHOH OKPY>KHOCTH TpeyrOJIbHH-
Ka AIBI C 1 HO HMeHHO 3TO Ii Tpe6yeTcSI )!OK333Tb. TeM caMblM, Mbl
CBeJIH CJIYLJaH TynoyroJlbHoro TpeyrOJIbHHKa ABC K CJIytl310 OCTpO-
yrOJIbHOrO TpeyrOJIbHHKa, 11 JlOKa33TeJIbCTBO Tpe6yeMoro yrBep>K.l1eHHH
nOJJHOCTblO 33BepllleHO.

188
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PemeHHH K  7
179. nYCTb Q - npoeKUHH TOlJKH P Ha CTO-
POHY AB, CM. PHCYHOK (TaK KaK P ueHTp
BnHcaHHOH OKPY>KHOCTH, TO PQ r). Tor1l3
113 npHMoyrOJIbHOrO TpeyrOJIbHHKa APQ HMe-
eM: r Ap. sin(Qj2) (*).
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no Teop e Me CH HYCOB H3 r p eyrOJIbHHKa ABA! HMeeM:
SIn a
npaBbIe qaCTH paBeHcTB (*) H (**), nOJlyqaeM: 2R. rAP · PAl, 4TO
H Tpe6oBanoCb ,1!OKa33Tb.
180. nepenHWM <jJOPMYJIY 3HJIepa B BH.l1e: 2R · r R 2 - d 2 . Boc-
nOJlb30BaBlllHCb pe3YJlbT3TOM npeJlblllYLUeH 3a,na4H, HMeeM: 2 R · r
AP · PAl, flle TOllKa P ueHTp BnHc3HHoH OKPY>KHOCTH IlaHHO-
ro TpeyrOJIbHHKa ABC, Al - TOllKa nepeCeqeHHH npSlMoH AP C onH-
caHHOH OKPY>KHOCTblO. fl03TOMY llJIH !lOKa33TeJlbCTBa <jJOPMYJIbI 3HJle-
pa ,llOCT3TOtIHO D-OKa3aTb, liTO Ap. P Al R 2 - d 2 . npOBeJleM llepe3
TOllKY P JlHaMeTp EF i'i 3aMeTHM, liTO AP · P Al EP · P F (H3BeCT-
Hoe CBOHCTBO ltBYX Xop.n. OKPY*HOCTH, CM. 3allalJY NQ37). 5e3 orpaHH-
qeHHH 06IUHOCTH CLJHTaeM, LJTO TOqKa P pacnOJlO)KeHa Ha palluyce DE,
cM.llaHHblH HH)Ke PHCYHOK. TOflla EP R - d, FP R + d (d - 3TO
u u u
paccToHHHe Me>KJIY ueHTpaMH BnHcaHHOH H onHcaHHOH OKPY)I{HOCTeH,
T.e.OP d) H,3H84HT, AP .PAI Ep.PF R 2 - d 2 , 4.T.,Q.
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OTBeTbl K 3aJ1.8lfHHKY
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189
OTBeTbl K 3aaqHHKY.
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1. 4 2. 9 3. 21 4. 35 5. 11 HJIH 79 - .naJIee BMeCTO COlO3a «HJ1H»
craBHM3HaK«;» 6.67.508.3;219.3010.24;5411.25;59
12.92 13.38 14.32 15.24 16.48 17.65 18.72 19.12 20.15
21. 20; 44 22. 3; 33 23.6; 24 24.21; 77 25. 12,5; 50 26. 10; 60
27. 18 28. 20 29. 17 30. 16 31. 22 32. 19 33. 100 iJ4. 13
35.25;165 36.70;130 37.30;150 38.50;110 39.140 40.90;170
41.165 42.108 43.80 45.129 46.64 47.25 48.150 49.35;55
50.26 51. 75; 150 52. 28 53.27 54. 45 55. 55 56.40 57. 130
58. 60 59. 30; 170; 10; 150 60. 60; 100; 40; 160 61. 8; 12 62. 23
63.164.5 65.14;58 66.32;48 67.12;100 68.16;30 69.7;43
70.12;38 71.16;44 72.3 73.3774.4;12;20;3675.6;48;54
76.31 77.41 78.5298 79.2448 80.10 81.19,32 82.4;6;7 83.5
84.11 85. a) 29; 6) n.(n1)+2 86.4; 5; 9 87.5; 6; 11 88.13 89.6
90.45 91.5 92.10 93.15 94. a)9; 6) n-(111-3)/2 95.HeT 96.7
97.8 98.9
2. TeopeMblo nape napaJIJleJlbHblX npHMblX H ceKYllI.eH
1.150 2.47 3.120 4.144 5.45 6.65 7.135 8.36 9.50 10.55
11.40 12.30 13.100
 3. TpeyroJlbHHK
18.99 19.10 20.115 21.68 22.36 23.90 24.44 25.72 26.102
27.8 28.80;65 29.40;55 30.100 31.64 32.45 33.7 34.110
35.152 36.78 37.75 38.118 39.77 40.105 41.20 42.70 43.30
44.15 45.50 46.95 47.73 48.14 49.120 54.28 55. 33 56.40

190
OTBeTbl K 38J1.8 l IHJfKY
it
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1. 495 2. 520 3. 4,5 4. 1,5 5. 9 6. 1 3,6 7. 3 8. 108 9. 54
-p p-
a q + b E- 8 a - l 9 P + 1 20
17. + 1. l b 1 · a) 6 : 5; 6) · a) 13 : 8;
p q - q
6) q + 1
p.q - 1
5. TeopeMa nH4Jaropa. CHHYC, KOCHHYC, TaHreHC
1. 13 2. 41 3. 15 4. 56 5. 14,5 6. 10 7. 25 8. 52 9. 24 10. 80
11. 37 12. 9 13. 7 4 t 4. 16 15. 9,8 16. 48 1 7. 7 18. 0,5 19. 1, 05
20. 120 21. 21 22. 14 23. 35J2 24. 3 34 25. 29 26. 5 27. 20
28.26 29.4V2 30. 1,6 33. 0,4 34.0,6 35. 1,5 36. 0,8 37. 1,5
38. 0,5 39. 3 40. 2,5 41. 2,4 42. 0,2 43. 44,8 44. 6 45. 1,875
46. 0,4 47. 12 48. 4 49. 36 50. 17,5 51. 10,5 52. 3,5 53. 16,5
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1.12,5 2.7,5 3.12 4.15 5.5,5 6.4,5 7.11 8. 10,5 9.6 10.9
1 t. 13,5 12. 14 13. 6 14. 5 15. 13 16. 10 17. 54 18. 60 19. 112,5
20. 42 21. 36 22. 8 23. 30 24. 150 25. 15; 75 26. 45 27. 168
28. 240 29. 34 30. 37 31. 126 32. 71,5 33. 18 34. 22 35. 116
36.720 37.25 38.56 39.46 40.62 41.189 42.17 43.32 44.16
45.29 46.140 47.175 48.390 49.10,8 50.122,5 51.21 52.27,5
53. 19 54. 64 55. 20 56. 48,6 57. 22,5 58. 26 59. 28 60. 231
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OTBeTbl K 3aJ/.allHJfKY 191
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