Геометрия7_9
.pdf8 KJiac |
90 |
ID;O H Tpe6a 6yno JJ.OBeCTH.
,ll;oBeJJ,eMO TOOpeMy JI.RH. aaranLHOro BHIIaJJ,KY (He JI.RH. aa11aM'H.-
TOByBaHHH.)• IIpH11ycTHMO, |
AC1 |
AB1 |
Ha11pHKnaJJ, |
AC1 |
> |
AB 1 |
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AC |
=I= AB , |
AC |
AB . |
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BiAKnaJJ,eMo Ha 11poMeHi AC BiApiaoK AC2 = |
~ ·AB, (Man. |
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139). IIpH ~oMy AC2 <AC,. Poai6'eMoBiJJ.piaoK AC Ha BenHKY KinLKiCTb n piBHHX qacTHH i IlpOBeJJ,eMO qepe3 roqKH noJJ,iny nps.Mi, napanenLHi BC.
IIpH JJ.OCHTL BenHKOMY n Ha BiApiaKy C1C2 6yJJ,yTL ToqKH 110JJ,iny. IIooHaqHMo OJI.HY a HHX qepea Y, a BiAnoBiJJ.HY ToqKy Ha BiApiaKy AB, qepea X. HK 6yno JJ.OBeJJ,eHo,
AY AX
AC= AB 0
3aMiHHMO y ajH piBHOCTi BenHqHHy AY Ha MeHmy BenHqHHY AC2, a BenHqHHY AX - Ha 6inLmy AB,. ,ll;icTaHeMo:
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AC2 < AB1 |
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AC |
AB 0 |
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AC |
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AC |
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3BlJJ.CH AC2 < |
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AB • AB,. |
Ane AC2 = AB • AB,. |
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MH npHHmRH JJ.O cynepeqHocTi. TeopeMy JJ.OBeJJ,eHo. |
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61. |
IIOBY,ll,OBA 'IETBEP'I'Oro |
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IIPOIIOPD;IHBOro Bl,ll,PI3KA |
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3 a A a q a |
6.1. lfaHo |
BiiJpiaJCu a, b, c. Ho6yiJyBaTu BiiJpi3oK |
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be |
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X= - . |
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a |
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Po a B's.3 a H H s.. |
ByJJ,yeMO |
JJ.OBiRLHHH HepooropHyTHH KYT |
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a BepmHHOIO 0 |
(Man. 140). BiAKnaJJ,aeMo Ha OJJ.HiH CTopoHi KyTa |
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BiJJ,piaKH OA = |
a i OB = |
b, Ha JJ,pyriii cropoHi - BiJJ,piaoK OC = |
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= c. C11onyqaeMo ToqKH Ai C nps.MOIO i qepea ToqKy B npoBOJJ.HMO napanenLHY i'H 11ps.My BD. BiApiaoK OD = x.
0
Man. 140
§ 6. qOTHpHKyTHHKH |
91 |
CnpaB,u;i, aa TeopeMoIO npo nponopn;i:iiHi Bi,u;piaKH
OA OC
OB= ov· |
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OD = OB · OC = |
be. |
OA |
a |
TaKHM 'IHHOM,Bi,u;piaoK OD 6 myKaHHM Bi,u;pi3KoM x. |
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3 a y Ba :>Ke H H SI. Ilo6y,u;oBaHHH |
Bi,u;piaoK x HaaHBa6ThCH |
'l.eree]JTUM nponop"iUHUM. u;a HaaBa ayMOBJieHa THM, :m;o BiH 6
'leTBepTHM'IJieHOMnponopn;fi a : b = c : x.
?
1•.
2.HKi BepmHHH 'IOTHpHKYTHHKaHa3HBaIOThCH cyci,u;HiMH? HKi HaaHBaIOThCH npOTHJie:>KHHMH?
3.mo TaKe ,u;iaroHaJii 'IOTHPHKYTHHKa?
4.HKi CTopoHH 'IOTHPHKYTHHKa HaaHBaIOThCH cyci,u;HiMH? HKi Ha3HBaIOThCH npOTHJie:>KHHMH?
5.HK noaHa11a6'J.'hCH'IOTHPHKYTHHK?
6.mo TaKe napaJieJIOl'paM?
7.AoBe,u;iTh, :m;o KOJIH ,u;iaroHaJii 'IOTHPHKYTHHKanepeTHHaIOThCH i B TO'ID;inepeTHHY ,u;iJIHThCH IlOilOJiaM, TO BiH 6 napaJieJiorpaMOM.
8.AoBe,u;iTh, :m;o ,u;iaroHaJii napaJieJiorpaMa nepeTHHaIOThCH i B TO'ln;inepeTHHY ,u;iJIHThCH IlOilOJiaM.
9.AoBe,u;iTh, :m;o y napaJieJiorpaMi npoTHJie:>KHi cropoHH piBHi, npoTHJie:>KHi KYTH piBHi.
10.mo TaKe Ilp.HMOKYTHHK?
11.AoBe,u;iTh, :m;o ,u;iaroHaJii npHMOKYTHHKa piBHi.
12.mo TaKe POM6?
13.AoBe,u;iTh, :m;o ,u;iaroHaJii poM6a nepeTHHaIOThCH ni,u; np.HMHM
KyToM; ,u;iaroHaJii poM6a 6 6iceKTpHcaMH iioro KYTiB.
14.mo TaKe KBa,u;paT? IlepeJii'liThBJiaCTHBOCTi KBa,u;paTa.
15.AoBe,u;iTh TeopeMy <l>aJieca.
16.AoBe,u;iTh, :m;o cepe,u;Ha JiiHia TPHKYTHHKa ,u;opiBHI06 noJIOBHHi
Bi,u;ITOBi,u;HOl CTOpOHH.
17.HKHii 'IOTHPHKYTHHKHa3HBa6ThCH Tpanen;i610?
18.HKa Tpanen;ia HaaHBa6ThCH piBHo6i11Ho10?
19.AoBe,u;iTh, :m;o cepe,u;HH JiiHia Tpanen;ii ,u;opiBHI06 niBcyMi ocHoB.
20.AoBe,u;iTh TeopeMy npo nponopn;iiiHi Bi,u;piaKH.
• |
3AM'II |
1.Ha MaJIIOHKax 114-116 no,u;aHo TPH cl>irypH, KO:>KHa a HKHX CKJia,ll;a6ThCH 3 'IOTMphOXTO'IOKi 'IOTHphOXBi,u;piaKiB, :m;o noCJii,ll;OBHO cnoJiy11a10Th n;i TO'IKH. HKa a n;Hx «Piryp 6 110THpHKyTHHKOM?
8 KJIRC |
92 |
2.Ilo6y,n;y:HTe µ;oBin:&HHH 'IOTHPHKYTHHKRQRS. IloKaJKiT& :Horo npOTHJieJKHi CTOpOHH Ta sepmHHH.
3.CKiJI&KH MOJKHa no6yµ;ysaTH napaJienorpaMiB a BepmHHaMH y TpbOX µ;aHHX TO'IKax,HKi He JieJKaTb Ha O,ll;HiH npSIMiH? Ilo6yµ;y:HTe ix.
4. Bi'IHa cTopoHa piBH06e,11;peHoro TPHKyTHHKa µ;opiBHIOs 5 M. 3 TO'IKH,B3$1TOl Ha OCHOBi ~bOl'O TPHKYTHHKa, npoueµ;eHo ,D;Bi npHMi, napaJieJI&Hi 6i'IHHMCTOpoHaM. 3HaH,n;iTb nepHMeTp yTBOpeHoro napaJieJiorpaMa.
5.Bi.n;cTaH& si,n; TO'IKHnepeTHHY µ;iaroHaJie:H napaJieJiorpaMa ,11;0 ,ll;BOX :Horo uepmHH µ;opiBHIOe 3 CM i 4 CM. qOMY µ;opiBHI06 Bi,D;CT8Hb Bi,n; Hei ,11;0 ,ll;BOX iHmHX BepmHH? IIOHCHiTb Bi,D;- noBi,D;b.
6.qepea TO'IKYnepeTHHY .n;iaroHaJie:H napaJieJiorpaMa npoueµ;eHo npHMy. ,IJ;oueµ;iT&, m;o Bi,n;piaoK ~isi npHMoi, HKHH aHaXOAHTbCH Mim napaJieJibHHMH CTOpOHaMH, ,n;in:HTbCSI B ajii TO'laj nonoJiaM.
7.Y napaJieJiorpaMi ABCD qepea TO'IKYnepeTHHY µ;iaroHaJieii
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npoueµ;eHo npHMY, HKa ui,n;THHas Ha cTopoHax BC i AD uiµ;pi3KH |
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BE= 2 Mi AF= 2,8 M. 3Ha:Hµ;iT& CTopoHH BC i AD. |
8. |
Y napaJieJiorpaMi ABCD AB = 10 CM, BC = 15 CM. qoMy |
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µ;opiBHIOIOTb cTopoHH AD i CD? IloHcHiT& ui,11;n0Bi,11;&. |
9. |
Y napaJieJiorpaMi ABCD L A = 30°. qoMy µ;opiBHIOIOTb KYTH |
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B, C, D? IloHCHiT& eiµ;noui,11;&. |
10.IlepuMeTp napaJieJiorpaMa ABCD µ;opiBHIOe 10 CM. 3Haiiµ;iT& AOBJKHHY µ;iaroHaJii BD, aHaIO'IH,m;o nepuMeTp TPHKYTHHKa ABD µ;opiBHI06 8 CM.
11.Oµ;HH a KYTiB napaJieJiorpaMa µ;opiBHIOe 40°. 3Haiiµ;iT& iHmi
KYTH.
12.3Haiiµ;iTb KYTH napaJieJiorpaMa, 3HaIO'IH,m;o O,ll;HH 3 HHX 6iJibmHii Bi,n; µ;pyroro Ha 50°.
13.qH MOJKe O,ll;HH 3 KyTiB napaJieJiorpaMa µ;opiBHIOB8TH 40°, a µ;pyrHii -- 50°?
14.,IJ;iaroHaJI& napaJieJiorpaMa yTsop10s a ,ll;BOMa A:oro cropoHaMH KYTH 25° i 35°. 3Haiiµ;iT& KYTH napaJieJiorpaMa.
15.3Haiiµ;iTb yci KYTH napaJieJiorpaMa, HKID;O cyMa ,ll;BOX ia HHX ,D;OpiBHI06: 1) 80°; 2) 100°; 3) 160°.
16.3HaH,D;iTb yci KYTH napaJieJiorpaMa, HKID;O piaHH~H ,ll;BOX ia HHX ,D;OpiBHI06: 1) 70°; 2) 110°; 3) 140°.
17. |
Y napaJieJiorpaMi ABCD TO'!Ka E - cepe,n;HHa cropoHH BC, |
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a F - |
cepe,n;uHa cTopoHH AD. ,ll;oBeµ;iT&, m;o 'IOTHpHKYTHHK |
18. |
BEDF - |
napaJieJiorpaM. |
,IJ;oueµ;iTb, m;o KOJIH y 'IOTHpHKYTHHKY,D;Bi CTOpoHH napaJieJibHi |
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ipiBHi, TO BiH 6 napaJieJiorpaMOM.
19.Y napaJieJiorpaMi ABCD npoueµ;eHo 6iceKTpHcy KyTa A, HKa
nepeTHHas c'fopoHyBC y TO'lajE. qoMy .n;opiBHIOIOTb Biµ;piaKH BE i EC, HKID;O AB = 9 CM, AD = 15 CM?
§ 6. qOTHpHKYTHHKH |
93 |
20.,ll;Bi CTOpOHH napammorpaMa Bi,ll;ltOC.STLC.S, .SK 3 : 4, a nepHMeTp iioro ,n;opiBHI06 2,8 M. 3Haii,n;iTL CTOpOHH.
21.Y napaJieJiorpaMi ABCD nepneH,n;HKyJiap, onym;eHHii a BepmHHH B Ha cTopoHy AD, ,n;iJIHTL ii'nonoJiaM. 3Haii,n;iTL ,n;iaroHaJIL BD i cTopoHH napaneJiorpaMa, KOJIH Bi.,n;oMo, m;o nepHMeTp napaJieJiorpaMa ,n;opiBHIOe 3,8 M, a nepHMeTp TPHKYTHHKa ABD p;opiBHI06 3M.
22.Ilo6y,n;yiiTe napaJieJiorpaM: 1) aa ,n;BoMa cTopoHaMH i ,n;iaroHaJIJI10; 2) aa CTOpOHOIO i ,ll;BOMa ,n;iaroHaJI.SMH.
23.IIo6y,n;yiiTe napaJieJiorpaM: 1) aa ,n;BoMa cTopoHaMH i KyToM;
2)aa ,n;iaroHaJI.&MH i KYTOM Mim HHMH.
24.,ll;oBe,n;iTh, m;o KOJIH y napaJieJiorpaMi BCi KYTH piBHi, TO BiH 6 np.SMOKYTHHKOM.
25.,ll;oBe,n;iTL, m;o KOJIH y napaJieJiorpaMi xoqa 6 O,ll;HH KYT np.&MHH, TO BiH 6 IlpSIMOKYTHHKOM.
26.,ll;oBe,n;iTL, m;o KOJIH y napaJieJiorpaMi ,n;iaroHaJii piBHi, TO BiH e npSIMOKYTHHKOM.
27.BeTOHHa IlJIHTa a nps:MoJiiHiiiHHMH CTOpoHaMH noBHHHa MaTH cl>opMy npaMOKYTHHKa. SIK aa ,n;on0Moro10 MOTY3KH nepeBipHTH npaBHJILHiCTL cl>opMH IlJIHTH?
28.BiceKTpHca o,n;Horo a KyTiB npaMOKYTHHKa ,n;iJIHTL iioro cTopoHy nonoJiaM. 3Haii,n;iTL nepHMeTp np.SMOKYTHHKa, .&Km;o iioro MeHma CTOpOHa ,n;opiBHIOe 10 CM.
29.y np.&MOKYTHHKY TO'IKa nepeTHHY ,n;iaroHaJieii 3HaXO,ll;HTLC.S Bi.,n; MeHmoi'CTOpoHH Ha 4 CM ,n;aJii, Him Bi.,n; 6iJILmoi'CTOPOHH. IlepHMeTp npaMOKYTHHKa ,n;opiBHI06 56 CM. 3Haii,n;iTL CTOPOHH np.SMOKYTHHKa.
30.3 o,n;Hiei'TO'IKHKOJia npoBe,n;eHO ,n;Bi B3a6MHO nepneH,ll;HKYJI.SpHi xop,n;H, Bi,n;,n;aJieHi Bi,n; ~eHTpa Ha 6 CM i 10 CM. 3Haii,n;iTL i'x ,ll;OBJKHHH.
31.y np.&MOKYTHHH TpHKYTHHK, KOJKHHH KaTeT .&Koro ,n;opiBHI06 6 CM, BilHCaHO np.&MOKYTHHK, SIKHH Mae 3 TPHKYTHHKOM cniJILHHH KYT (MaJI. 141). 3HaH.n;iTL nepnMeTp npaMOKYTHHKa.
32.y piBH06e,n;peHHH npaMOKYTHHH TpHKYTHHK BilHCaHO npaMO-
KYTHHK T&K, |
m;o ,n;Bi iioro BepmHHH JiemaTL Ha rinoTeHyai, |
a ,ll;Bi iHmi - |
Ha KaTeTax (MaJI. 142). 'loMy,n;opiBHIOIOTL CTO- |
Man. 141 |
Man. 142 |
8 KJiaC |
94 |
poHH npSIMOKYTHHKa, KOJIH Bi,ZJ;OMO, m;o BOHH Bi,ZJ;HOCSITLCR, SIK 5 : 2, a rinoTeHyaa TpHKYTHHKa µ;opiBHI06 45 CM?
33.,lJ;oBe,ZJ;iTL, m;o KOJIH y napaJieJiorpaMi ,ZJ;iaroHaJii nepneH,ZJ;HKyJISlpHi, TO BiH 6 poM60M.
34.,lJ;oBeµ;iTL, m;o KOJIH µ;iaroHaJIL napaJieJiorpaMa e 6iceKTpHco10 iioro KYTiB, TO BiH 6 poMOOM.
35.KyTH, YTBopeHi µ;iaroHaJISIMH poM6a a oµ;Hi610 a :Doro cTopiH, Bi,ZJ;HOCSITLCSI, SIK 4 : 5. 3Haii,ZJ;iTL KYTH poM6a.
36.,lJ;oBe,ZJ;iTL, m;o 'IOTHpHKYTHHK, y SIKOro BCi CTOPOHH piBHi, 6 poM60M.
37.y poM6i O,ZJ;Ha 3 µ;iarOHaJieH ,ZJ;OpiBHI06 CTOPOHi. 3HaH,ZJ;iTL KYTH poM6a.
38.Ilo6yµ;y:HTe poM6: 1) aa KYTOM i .n.iaroHaJIJIIO, mca BHXO,ZJ;HTb 3 BepIIIHHH ~Loro KyTa; 2) aa ,ZJ;iaroHaJIJIIO i npoTHJiemHHM KyTOM.
39.Ilo6y,ZJ;yHTe poM6: 1) aa CTOpOHOIO i µ;iaroHaJIJIIO; 2) aa ,ll;BOMa ,ZJ;iaroHaJISIMH.
40.,lJ;oBe,ZJ;iTL, m;o KOJIH ,ZJ;iaroHaJii npSIMOKYTHHKa nepeTHHaIOTbCSI ni,ZJ; npSIMHM KYTOM, TO BiH e KBa,ZJ;paTOM.
41.y piBH06eµ;peHHH npSIMOKYTHHH TPHKYTHHK, KomeH KaTeT SIKOro 2 M, BilHCaHO KBa,ZJ;paT, m;o Ma6 3 HHM cniJILHHH KyT. 3Haii- ,ZJ;iTL nepHMeTp KBa,ZJ;paTa.
42.,lJ;aHo KBaµ;paT ABCD. Ha KOmHiH a iioro cTopiH Bi,ZJ;KJia,ZJ;eHo piBHi Biµ;piaKH: AA1 = BB1 = CC1 =DD,. ,lJ;oBe,ZJ;iTL, m;o 110THpHKYTHHK A1B1C1D1 - KBa,ZJ;paT (MaJI. 143).
43.,IJ;iaroHaJIL KBa,ZJ;paTa ,ZJ;opiBHIOe 4 M. CTopoHa iioro µ;opiBHI06 ,ZJ;iaroHaJii µ;pyroro KBa,n;paTa. 3aa:H,ZJ;iTL cTopoHy µ;pyroro KBa,ZJ;- paTa.
44.,IJ;aao KBa,ZJ;paT, cTopoaa .RKoro 1 M, µ;iaroaaJIL :Doro .n;opiBHIOe cTopoai ,ZJ;pyroro KBa,ZJ;paTa. 3aa:H,ZJ;iTL µ;iaroaaJIL ,ZJ;pyroro KBaµ;paTa.
45.y KBa,ZJ;paT BilHCaHO npSIMOKYTHHK TaK, m;o Ha KOmHiH CTOpoHi KBa,ZJ;paTa 3HaXO,Zl;HTLCSI O,ZJ;Ha BepmHHa npSIMOKyTHHKa i CTOpo-
A D
[,
B
MaJI. 143 |
MaJI. 144 |
§ 6. qOTHpHKYTHHKH |
95 |
HHIIp.HMOKYTHHKa napaJieJihHi ,D;iaroHaJI.HM KBa,D;paTa (MaJI.
144). 3HaH,D;iTb CTOpOHH np.HMOKYTHHKa, 3HaIOqH, m;o O,ll;Ha 3 HHX Y,D;Biqi 6iJibIDa Bi,ll; ,D;pyro'i,a ,D;iaroHaJib KBa,D;paTa ,ll;OpiBHI06 12 M.
46.y piBH06e,D;peHHH np.HMOKYTHHH TpHKYTHHK BIIHCaHO KBa,D;paT
TaK, m;o ,ll;Bi iioro BepmHHH 3HaXO,ll;.HTbC.H Ha rinoTeHyai, a ,ll;Bi iHmi - Ha KaTeTax. 3HaH,D;iTb CTOpOHY KBa,D;paTa, KOJIH Bi,ll;OMO, m;o rinoTeHyaa ,D;opiBHIOe 3 M.
47.3 ,D;aHo'iTOqKH npoBe,D;eHO ,ll;O KOJia pa,ll;iycoM 10 CM ,ll;Bi B3a6MHO nepneH,D;HKYJI.HpHi ,ll;OTffqHi. 3Haii,ll;iTb ,ll;OB»CHHH ,ll;OTHqHHX (Bi,ll;CTaHb Bi,ll; ,D;aHo'iToqKH ,ll;O ToqKH ,ll;OTHKy).
48.IIo,ll;iJiiTb ,D;aHHH Bi,D;piaoK AB Ha n piBHHX qacTHH.
49.Ilo,D;iJiiTb ,D;aHHii Bi,D;piaoK Ha TaKy KiJihKiCTb piBHHX qacTHH:
1)3; 2) 5; 3) 6.
50.CTopoHH TPHKYTHHKa ,D;opiBHIOIOTb 8 CM, 10 CM, 12 CM. 3Haii-
,D;iTh CTOPOHH TPHKYTHHKa, BepmHHaMH .HKoro 6 cepe,D;HHH CTOpiH ,D;aHoro TpHKYTHHKa.
51.IlepHMeTp TpHKYTHHKa ,ll;OpiBHI06 12 M, cepe,D;HHH CTOpiH CIIOJiyqeHO Bi,D;piaKaMH. 3Haii,D;iTh nepHMeTp yTsopeHoro TPHKYTHHKa.
52.Cepe,D;HB JiiHia piBHo6e,D;peHoro TpHKYTHHKa, napaJieJihHa ocHoBi, ,ll;OpiBHI06 3 CM. 3HaH,D;iTb CTOpOHH TPHKYTHHKa, .HKIIl;O iioro nepHMeTp CTaHOBHTb 16 CM.
53.HK no6y,D;yBaTH TPHKYTHHK, .aKm;o ,D;aHo cepe,D;HHH iioro cTopiH?
54.,ll;oBe,D;iTh, m;o BepmHHH TpHKYTHHKa piBHOBi,D;,D;aJieHi Bi,ll; np.a- MO'i,BKa npOXO,ll;HTb qepea cepe,D;HHH ,ll;BOX iioro CTOpiH.
55.,ll;oBe,D;iTh, m;o cepe,D;HHH CTOpiH qoTHpHKYTHHKa 6 BepmHHaMH napaJieJiorpaMa.
56.3Haii,D;iTh CTopoHH napaJieJiorpaMa a nonepe,D;Hbo'i aa,D;aqi,
HKIIl;O ,ll;iaroHaJii qoTHpHKYTHHKa ,ll;OpiBHIOIOTb 10 M i 12 M. 57. y qoTHPHKYTHHKY ,D;iaroHaJii ,lJ;OpiBHIOIOTb a i b. 3HaH,D;iTb nepHMeTp 'IOTHpHKYTHHKa, BepmHHaMH HKOro 6 cepe,D;HHH CTOpiH
,D;aHoro 'IOTHPHKYTHHKa.
58.,ll;oBe,D;iTh, m;o cepe,D;HHH CTOpiH npaMOKYTHHKa 6 BepmHHaMH poM6a. I HasnaKH, cepe,D;HHH cTopiH poM6a e sepmHHaMH IIpBMOKYTHHKa.
59.Bi'IHYCTOpOHY Tpane~'i IIO,ll;iJieHO Ha TPH piBHi 'laCTHHH i
qepea TO'IKHIIO,ll;iJiy npoBe,D;eHO ,ll;O ,D;pyro'iCTOpoHH Bi,D;piaKH, napaJieJihHi OCHOBaM. 3HaH,ll;iTb ,ll;OB»CHHH ~x Bi,D;piaKiB, HKIIl;O OCHOBH Tpane~'i ,ll;OpiBHIOIOTb 2 M i 5 M.
60.,ll;oBe,D;iTh, m;o B piBHOOi'IHiHTpane~'i KYTH IIPH OCHOBi piBHi.
61.qOMY ,lJ;OpiBHIOIOTb KYTH piBHOOi'IHOlTpane~'i. HKIIl;O pi3HH~.H
npOTHJie»CHHX 'i'iKyTiB ,ll;OpiBHI06 40°?
62.y piBH06i'IHiHTpane~i'i 6iJibma OCHOBa ,ll;OpiBHI06 2,7 M, 6i-•rna CTOpOHa 1 M, a KYT Mim HHMH 60°. 3HaH,D;iTb MeHmy OCHOBy.
63.Y pisHo6i'IHiHTpane~'i BHCOTa, npose,D;eHa a sepmHHH Tynoro
8 Knac
8
65.3 OAHOro 6oRy Bi.ii; npaMoi a AaHo ABi TO'IKHA i B Ha Bi.zi;cTaHi 10 Mi 20 M Bi.ii; He'i.3Haii.a;iTb Bi.zi;CTaHb Bi.ii; cepeAHHH SiAPi3Ka AB AO npaMo'ia.
66.3 piaHHX 6oRiB Bi.ii; npaMo'ia .a;auo .a;si TO'IKHA i B Ha Bi.zi;cTaHsa 10 CM i 4 CM BiA He'i.3Haii.a;iTb si.zi;cTaHb sip; cepe.a;HHH BiApiaKa AB AO npaMo'ia.
67.OcHosH TPane~'i Bi.zi;HocwrDCa, HK 2 : 3, a cepe.a;Ha JiiHia AOPiB-
Htoe 5 M. 3Haii.a;iTb OCHOBH.
68. KiH~ .a;iaMeTpa sip;p;aJieHi |
BiA AOTH'IHO'iAO KOJia Ha 1,6 M i |
0,6 M. 3Ha:ii.a;iTb AOBJKHHY |
.a;iaMeTPa. |
69.CepeAHH JiiHiH Tpane~'i .a;opiBHI06 7 CM, a O,ll;Ha 3 'i'i OCHOB 6iJihma sip; APYro'iHa 4 CM. 3Haii.a;iTb OCHOBH Tpane~'i.
70.BHcOTa, nposeAeHa a BepmHHH TYnoro KyTa piBHo6i'IHO'iTpane- ~ii, AiJIHTb 6iJibmy OCHOBY Ha 'laCTHHH,~o MalOTb ,ll;OBatHHH a i b (a> b). 3Haii,ll;iTb cepe.a;HIO niHi10 Tpane~'i.
71tr Ilo6YAYHTe |
Tpane~iIO aa OCHOBaMH i 6i'IHHMH CTOpOHaMH. |
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72'!'Ilo6y,n;yiiTe Tpane~10 aa OCHosaMH i AiaroHaJIHMH. |
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73tr ,Il;aHo |
Bip;piaKH a, b, c, d, e. Ilo6y.a;yiiTe sip;piaoK x = a::. |
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74tr 1) Y |
TPHKYTHHKY ABC npoBe,n;eHo MeAiaHH AA, e BBi. HKi |
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nepeTHHatoTDCH B TO'I~ M (Man. 146). Y TPHKYTHHKY AMB |
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npoBe,n;eHo cepe,11;H10 JiiHiIO PQ. ,Il;oBeAiTb, ~o 'IOTHPHKYTHHK |
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2),IJ;ose,ll;iTb, ~o ,D;Bi AOSiJibHi Me,n;iaHH TpHKYTHHKa TO'IKOIO nepeTHHY AiJISTbCJI y Bi.zi;HomeHHi 2 : 1, IlO'IHHalO'IHBi.ii; BepWHHH.
3),Il;oBe.a;iTb, ~o sci TPH Me.a;iaHH TPHKYTHHKa nepeTHHaIOTbCH B OAHi~ TO'I~.
§ 7 . TeopeMa Ilicl>aropa |
97 |
§ 7. TEOPEMA IlICl>ArOPA
62. KOCHHYC KYTA
KocuHycoM rocTporo Kyra npRMOKY'fHoroTPHKY'fHHKaHaaHBaGTbCR BiAHOWeHHR npHJiernoro KaTeTa AO rinoTeHy3H.
KocHHyc KyTa a noaHa11aeTbCR TaK: cos a. Ha MaJIJOHKY 147 ao6pameHo npRMOKYTHHH TPHKYTHHK ABC a KYTOM A, RKHH AopiBHJOe a. KocHHyc KyTa a AOpiBHJ06 BiAHomeHHJO KaTeTa AC, npHJiernoro AO u;Loro KyTa, AO rinoTeHyaH AB, To6To
AC
COS a= AB.
"°'Teo p e Ma 7.1. Kocunyc Kyra aaJU!:HCUT'b riabJCU BiiJ zpaiJyc11&ipu Kyra i 1U! aaJU!:HCUT'b BiiJ poa11du+enna i poaxipiB rpuKyT-
nuxa.
u;e 03H8'186, m;o y ABOX npRMOKYTHHX TPHKYTHHKiB 3 OAHHM i THM caMHM rocTpHM KYTOM KOCHHyCH IJ;bOro KyTa piBHi.
,IJ; o Be A e H HR. Hexa:H ABC i A'B'C' - ABa npRMOKyTHi TPHKYTHHKH a OAHHM i THM caMHM KYTOM npH BepmHHax A i A', RKHH AOPiBHJ06 a (MaJI. 148). Tpe6a AOBeCTH, m;o
A'C' AC
A'B' =AB"
Ilo6YAY6MO TPHKyTHHK AB1C1, RKHH AOPiBHIOe TPHKYTHHKY A'B'C' TaK, .RK noKaaaHo Ha MaJIJOHKY 148. OcKiJibKH np.RMi BC i B1C1 nepneHAHKYJIRpHi AO npRMoi'.AC, TO BOHH napaJieJibHi. 3a TeopeMOIO npo nponopu;iiiHi BiApiaKH M86MO:
AC1 AC
AB1 =AB"
A 11epea Te m;o aa no6yAOBOJO AC, = A'C', AB 1 = A'B', TO
A ' C' AC
A'B' =AB"
TeopeMy AOBeAeHo.
MaJI. 147 |
MaJI. 148 |
4 feoMOTpiH, 7-9 KJJ .
8 KJI8C |
98 |
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63. TEOPEMA DICZ-Al'OPA
Teo p e Ma 7.2 (TeopeMa Ilicl>aropa). Y npJU1Wxyr1W11&JI rpu- x yrHUxy 1ct1aiJpar zinoreHyau iJopitlHKle cJiJtd JUJaiJpari.8 xareritl.
,ll; o B e A e H H a. Hexa:A ABC - 11.amdi np11MoKyTHHii TPHKYTHHK a npRMHM KYTOM C. IlposeJJ.eMo BHCOTY CD a seplllllHH npRMoro KYT& C (MaJI.
[
A |
0 |
B |
Man. 149
AD AC
KOCHuyca KyTa cos A = AC = AB·
3si.AcH AB · AD = AC2• A uaJioriquo cos B = ~ = ~· 3BiJJ.CH
AB· BD = BC2• ,ll;o11.asmH piBuocTi noqneuuo i spaxysasmu, ~o AD + DB = AB, JJ.iCTaHeMo:
AC2 + BC2 = AB (AD + DB) = AB2•
TeopeMy 11.ose11.euo.
3 TeopeMB Ili<Paropa BHnJIHBae, ~o 11 npJ1MoxyTH011&y rpuxyrHUxy 6fiJ•rJIJCUU a xareritl Jl&eHIAUU 34 zinoreHyay. 3Bi.ACH, B CBOIO
qepry, BHilJIHBae, ~o coe ct < 1 AJ1J1 |
fiYAJ>-SKOl'OrocTpOro E}'Tact. |
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3 a 11. a q a |
(11). 3ua:AJJ.iTb Me11.iauy piBuo6e11.peuoro TPH- |
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KYTHHKa 3 OCHOBOIO a i 6iquo10 cropOHOIO b, npose11.euy 11.0 |
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OCHOBH. |
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P o a s'a a a H H a. |
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ABC - |
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piBHo6eJJ.peHHH TPBKYTHHK 3 |
OCHOBOIO |
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AB i CD - iioro MeJJ.iaua, npose11.eua p,o |
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OCHOBH (MaJI. 150). HK si.AoMo, MeJJ.iaua |
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piBuo6ep,peuoroTpHKyTHHKa,npose11.eua |
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11.0 OCHOBH, e BHCOTOIO. ToMy TPBKYTHHK |
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ACD npRMoKyTHHii a npm.tHM KYTOM D. |
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3a |
TeopeMOIO Ilicl>aropa AD2 + CD2 = |
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= |
AC2 , T 2 |
+ CD2 = |
b2. 3Bip,cuCD= |
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D |
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b2 - |
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Man. 150 |
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§ 7. TeopeMa Ili«Paropa |
99 |
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64. srHIIETChKHH |
TPHKYTHHK |
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0 |
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3 |
a A a 11 a |
(17). ,ZJ;oBe,ll;iTb, |
w;o KOJIH y |
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TPHKYTHHKY CTOpoHH a, b, c i a 2 |
+ b2 = |
c2, |
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TO KYT TPHKYTHHKa, npoTHJieJKHHH CTOpOHi |
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C, |
npSIMHH. |
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Po a B'a a a H H n. |
Hexaii ABC - |
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HHii TPHKYTHHK, y $1KOro AB = |
c, AC = |
a, |
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BC= b (MaJI. 151). Ilo6y,ll;yeMo npaMoKyT- |
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HHii TPHKYTHHK A 1B1C1aKaTeTaMHA1C1 = |
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= |
a |
i B1C1 = |
b. |
3a TeopeMoIO |
Ilid>aropa |
' |
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iioro |
rinoTeHyaa |
A1B1 = ../a2 |
+ b2 = |
c. |
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TaKHM 'IHHOM,TPHKYTHHKH ABC i A1B1C1 |
IIicl>arop - p;aBHbO- |
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piBHi aa TpeTbOIO 03HaKOIO. 3 piBHOCTi |
rpe~bKHH y<leHHH |
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TPHKYTHHKiB BHilJIHBae, w;o KYT TPHKYT- |
(VI CT . ,D;O H. e .) |
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HHKa ABC npH BepmHHi C npaMHii. |
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3eMJieMipH CTapo,!J;aBHboro 6rnnTy ,ll;JISI no6y,ll;OBH npaMoro KyTa KOPHCTYBaJIHCb TaKHM cnoco6oM. MoTyaoK ,ll;iJIHJIH ByaJiaMH Ha 12 piBHHX 11acTHH i KiH~ aB'SiayBaJIH.IIoTiM MOTyaoK poaTSiryBaJIH Ha aeMJii TaK, w;o6 YTBOpHBCSI TPHKYTHHK ia CTOpOHaMH 3, 4 i 5 IlO,ll;i-
JIOK. |
KyT TpHKYTHHKa, |
npoTHJieJKHHii ,ll;O cTopoHH, SIKa Mae 5 no,ll;i- |
JIOK, |
6yB npHMHH (32 |
+ 42 = 52). y 3B'H3KY 3 T8KHM cnoco6oM |
no6y,ll;OBH npSIMoro KyTa TPHKYTHHK ia cTopoHaMH 3, 4, 5 O,ll;. iHKOJIH HaaHBaIOTb ezunercbJCUM..
A A,
[ |
B c, |
B, |
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MaJI. 151 |
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65. IIEPIIEUWIKYJUIP I IIOXHJIA
Hexaii BA - nepneH,ll;HKyns:p, onyw;eHHii a TO'IKHB Ha npSIMY a, i C - , 6y,ll;L-$1Ka TO'll<anpSIMo1 a, Bi,ll;MiHHa Bi,ll; A. Bi,ll;piaoK BC HaaHBa6TbCH noxu.n.mo, npoBe,ll;eHoIO a TO'IKHB ,ll;O np.s:Mo1 a (MaJI. 152). To'IKaC HaaHBaeTbCSI ocnooo10 noxu.n.o'i. Bi,ll;piaoK AC HaaHBaeTbca npoe1e11tie10 noxuAoi.
3 TeopeMH 1Ii4>aropa BHilJIHBae, w;o XOJl.U a iJaHO'i T01'1CU iJo npsiMO'i npoBeiJeHo nepneHiluxyAsip i noxull.i, ro 6yiJ1>-sixa noxuAa