Геометрия7_9
.pdf9 KJiac |
170 |
A
Man. 255 |
Man. 256 |
34.qH IlOAi6Hi AB8 piBHOCTOpOHHi TPHKYTHHKH?
35.qH noAi6Hi TpHKYTHHKH ABC i A1B1C1, .RK~o:
1) |
AB= 1 M, AC= 1,5 M, BC= 2 M; A1B1 = |
10 CM, A1C1 = |
= |
15 CM, B1C1 = 20 CM; |
8 AM, A1C1 = |
2) |
AB= 1 M, AC= 2 M, BC= 1,5 M; A1B1 = |
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16 AM, B1C1 = 12 AM; |
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3) |
AB= 1 M,AC = 2 M,BC = 1,25 M; A1B1 = |
10 CM,A1C1= |
= |
20 CM, B1C1 = 13 CM? |
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36.,lJ;oBeAiTh, ~o y IlOAi6HHX TPHKYTHHKiB nepHMeTpH BiAHOCRTbCR, HK BiAnoBiAHi CTOPOHH.
37.CTopoHH TPHKYTHHKa AOpiBHIOIOTh 0,8 M, 1,6 Mi 2 M. 3HaHAiTh cTopoHH noAi6Horo HOMY TPHKYTHHKa, nepHMeTp SKoro AOPiBHIOG 5,5 M.
11
38. IlepHMeTp OAHOro TPHKYTHHKa CT8HOBHTb 13 nepHMeTpa
noAi6Horo HOMY TpHKYTHHKa. PiaHH~H ABOX BiAnoBiAHHX CTOpiH AOPiBHIOG 1 M. 3HaHAiTh ~ cropoHH.
39.qH IlOAi6Hi AB8 npHMOKYTHi TPHKYTHHKH, HK~O OAHH 3 HHX Ma6 KYT 40°, a APYrHii - KyT, ~o AOPiBHI<>e: 1) 50° ; 2) 60°?
40.OcHoBa BHCOTH npHMOKYTHOro TPHKYTHHKa, ony~eHo'i Ha rino-
TeHyay, AiJIHTb 'i'i Ha BiAPi3KH 9 CM i 16 CM. 3HaHAiTb CTOPOHH TPHKYTHHKa.
41. rinoTeHyaa npHMOKYTHOro TpHKYTHHKa AOpiBHI06 25 CM, a OAHH a KaTeTiB AopiBHI06 10 CM. 3HaHAiTh npoeK~iIO APYroro KaTeTa Ha rinoTeHyay.
42.,lJ;oBeAiTh, ~o BiAilOBiAHi BHCOTH IlOAi6HHX TPHKYTHHKiB BiAHOCHTbC$1, HK BiAilOBiAHi CTOpOHH.
43.KaTeTH npsMOKYTHoro TpHKYTHHKa BiAHOC8TbCH HK m : n. HK BiAHOCHTbCH npoeK~'i KaTeTiB Ha rinOTeHyay?
44.,lJ;oB.>KHHa TiHi cl>a6pH11Ho'iTpy6H AOPiBHIOG 35,8 M. Y TOH caMHH 11ac BepTHK~JlbHO IlOCT8BJieHa mepAHHa aaBAOB.>KKH 1,9 M Aa6
TiHb AOBJKHHOIO 1,62 M (MaJI. 257). 3H8HAiTb BHCOTY Tpy6H.
§ 11. IIo,l:\i6HicTL <Piryp |
171 |
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35,8 |
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MaJI. 257 |
MaJI. 258 |
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45. Y TPHKYTHHK ABC BrrHcaHo poM6 ADEF TaK, m;o KYT A y HHX crrinLHHii, a BepmHHa E JiemHTL Ha cTopoHi BC (MaJI. 258). 3Haii,11;iTL cTopoHy poM6a, aKm;o AB = c, AC = b.
46:'<BiceKTpHca aoBHiIIIHLoro KyTa TpHKYTHHKa ABC rrpH BepIIIHHi C rrepeTHHae rrpaMy AB B To11:cV D (MaJI. 259). ,D;oBe,D;iTL, m;o
AD : BD = AC : BC.
47"!',D;oBe,n;iTL, m;o reoMeTpH'IHHMMic~eM TO'IOK,Bi,ll;HOIIIeHH$1 Bi,ll;- CTaHeii Bi,ll; $1KHX ,ll;O ,ll;BOX ,n;aHHX TO'IOKCTaJie (He ,n;opiBHI06
O,ll;HHH:cV), |
6 KOJIO. |
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48. 3Haii,n;iTL |
,ll;OIIOBH$1JlhHi IIJIOCKi KYTH, |
8KIII;O: |
1) |
O,ll;HH 3 HHX y |
5 paa 6iJILIIIHii aa ,n;pyrHii; 2) O,ll;HH |
3 HHX |
Ha |
100° 6iJILIIIHH |
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aa,n;pyrHii; 3) piaHH~a ix ,11;opiBHI06 20°.
49.To11KH A, B, C JiemaTi. Ha KOJii. 1ioMy ,n;opiBHI06 xop,n;a AC, $1KIII;O KYT ABC ,n;opiBHI06 30°' a ,11;iaMeTp KOJia 10 CM?
50.To11KH A, B, C JiemaTL Ha KOJii. 1ioMy ,n;opiBHI06 KYT ABC, aKm;o
xop,n;a AC ,11;opiBHI06 pa,n;iycy KOJia? (,D;Ba BHrra,n;KH.)
51.,D;oBe,D;iTL, m;o ~eHTpOM KOJia, OIIHCaHoro HaBKOJIO rrpaMOKYTHOro TPHKYTHHKa, e cepe,n;HHa rirroTeHyaH.
52.,D;oBe,n;iTL, m;o Me,D;iaHa rrpaMOKYTHOro TpHKYTHHKa, rrpoBe,n;eHa ,n;o rirroTeHyaH, poa6HBa6 iioro Ha ,n;Ba piBH06e,11;peHHX TPHKYTHHKH.
53.IIo6y,n;yiiTe rrpaMOKYTHHH TPHKYTHHK aa ri110TeHyao10 i BHco-
TOIO, orrym;eHOIO 3 BepIIIHHH rrpaMoro KyTa Ha rirroTeHyay.
54.Ha KOJii 11oaHa11eHo 110THPH TO'IKH:A, B, C, D. 1ioMy ,n;opiBHI06 KYT ADC, aKm;o KYT ABC ,11;opiBHI06 a? (,D;Ba BHrra,n;KH.)
55.Xop,n;H KOJia AD i BC rrepeTHHaIOTLca. KyT ABC ,n;opiBHI06 50°,
a KYT ACD ,11;opiBHI06 80°. 3Haii,n;iTL KyT CAD.
56:'<,D;oBe,n;iTL, m;o y 'IOTHPHKYTHHKy,BIIHCaHOMY y KOJIO, cyMa rrpoTHJie)KHHX KyTiB ,n;opiBHI06 180°.
57.,D;oBe,n;iTL, m;o reoMeTpH'IHHMMic~eM BepIIIHH rrpaMHX KyTiB, CTOpOHH $1KHX rrpoxo,n;aTb 11epea ,ll;Bi ,n;aHi TO'IKH,6 KOJIO.
9 KJI8C |
172 |
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MaJI. 259 |
MaJI. 260 |
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MaJI. 261 |
MaJI. 262 |
58. |
,D;oBeAiTh, ~o reoMeTpH'IHHMMicqeM BepmHH KyTiB 3 ARHOIO |
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rpaAYCHOIO MipoIO, CTOPOHH $1KHX npoXOA$1Tb qepe3 ABi ARHi |
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TO'IKH,a sepIIIHHH JiemaTh 3 OAHoro 6oKy BiA npHMO'i,~o CilOJiy- |
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59. |
qae qi TO'IKH,6 AYra KOJia 3 KiHqHMH B ~HX TO'IKaX(MaJI. 260). |
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,D;oBeAiTh, ~o rocTpHH KYT Mim XOPAOIO KOJia i AOTH'IHOIOAO |
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KOJia B KiHqi xopAH AOPiBHIOe noJIOBHHi KyTa Mim paAiycaMH, |
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60. |
npoBeAeHHMH AO KiHqiB XOPAH (MaJI. 261). |
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Ilo6YAYHTe TpHKYTHHK aa CTOpOHOIO, npOTHJieatHHM 'iH KyTOM |
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61. |
i BHCOTOIO, npose,u;eHOIO 3 BepmHHH qi,oro KyTa. |
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3 TO'IKHC KOJia npoBeAeHo nepneH,ZJ;HKYJIHP CD ,u;o ,u;iaMeTpa |
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62. |
AB. ,D;oseAiTb, ~o CD2 = |
AD · BD. |
,D;ose,u;iTh, ~o ,u;o6yTOK Bi,u;piaKiB Ci'IHO'iKOJia ,u;opiBHI06 KBa,u;pa- |
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Ty si,u;piaKa ,u;oTH'IHo'i,npose,u;eHo'ia Tie'icaMoi TO'IKH: ACX |
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X BC = CD2 (MaJI. 262).
63. .HK AaJieKO BH,ZJ;HO 3 JiiTaKa, $1KHH JieTHTb Ha BHCOTi 4 KM HaA 3eMJieIO, HK~O pa,u;iyc 3eMJii 6370 KM?
64.06'1HCJiiTbpa,u;iyc ropH30HTy, ~o BH,ZJ;HO 3 BepmHHH TeJie6amTH B KH6Bi, BHCOTa HKOl 373 M.
§ 12. P03B 'Sl3Y BRH HSI TPHK YTHHKiB |
173 |
§ 12. P03B'H3YBAHHHTPHKYTHHKIB |
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109. TEOPEMA KOCHHYCIB
Teo p e Ma 12.1 (TeopeMa KOCHHyciB). KBaiJpaT 6yiJb-1uc.o'i CTopmiu TpUKYTHUKa iJopi81UOe cyMi 1'8aiJpaTi8 080X inmux CTOpin 6e3 noiJBoenoto iJo6yTKy q.ux CTOpiH Ha ICOCUHYC ICyTa Mi"' HUMU.
,n; o B e TI. e H H a. |
Hexaii ABC - |
TI.RHHH TPHKYTHHK (MaJI. 263). |
,D;oBell,eMo, ~o BC2 = |
AB2 + AC2 - |
2AB · AC · cos A. |
8 D |
A |
B |
a) |
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t5) |
Man. 263 |
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Man. 264 |
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MaeMo BeKTOPHY piBHiCTb BC = |
AC - AB. IliTI.HOCHMO u;10 piB- |
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HicTb CKaJiapHO AO KBall,paTa, ll,iCTaHeMo: |
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BC2 = |
AB2 + AC2 - 2 AB · AC, |
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a6o |
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BC2 = AB2 + AC2 - |
2 AB · AC · cos A . |
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TeopeMy TI.OBell,eHo.
9 KJI8C |
174 |
3ayBaatHMO, ~o AC · cos A AOPiBHI06 aa a6conIOTHOIO BenHqHHOIO npoeKn;i'iAD CTOpOHH AC Ha cTopoHy AB (Man. 263, a) a6o iI npoµ;oBmeHHSI (Man. 263, 6). 3HaK AC · cos A aaneatHTb Biµ; KYTaA: HK~o KYT A rocTpH:H, TO 6epeTbCSI •+ •, HK~o Tynu:H, TO •-•. 3BiACH Ma6MO HacniAoK: 1C6aiJpar CTOpOKU rpuicyTKUJCa iJopi6K10e cyMi 1C6aiJpari6 060X iKULUX CTopiK (( + » noiJBoeKUU iJo6yroK oOKie'i a KUX Ha npoe1C1fil0 iJpyzo'i. 3HaK « +» CJ&iiJ 6paTU ToOi, ICOJ&U nporuae31CKUU icyr rynuii., a aHaic «-»,- icoau zocrpuu.
. 3 a A a 11 a |
(7). ,lJ;aHo cTopoHH TPHKYTHHKa a, |
b, |
c. 3Haii- |
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,ll;1Tb BHCOTY TpHKYTHHKa, ony~eHy Ha CTOfOHY c. |
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Po a B'SI a a H HSI. |
Ma6MO |
a 2 |
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b |
+ c2 |
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2c ·AD |
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(Man. 264). |
3BiACH |
AD= + |
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a 2 - |
b 2 - |
c2 |
3a |
TeopeMOIO |
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Ili<Paropa |
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2 )2 |
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CD= .jAC2 - |
2 |
b 2 |
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AD2 = |
yb2 - |
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2c - |
c |
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110. TEOPEMA CHHYCIB
Teo p e Ma 12.2 (TeopeMa CHHyciB). CropoKu rpuicyrnu-,ca |
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nponop1fiUKi 00 CUKyci6 nporwie31CKUX icyTi6. |
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A o B e A e H H SI. Hexaii ABC - TPHKYTHHK ia cTopoHaMH a, |
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b, c i |
npOTHneatHHMH KyTaMH a., ~. 'V (Man. 265). ,lJ;oBe,u;eMO, ~o |
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b |
c |
sin a = |
sin fl = |
sin y • |
OnycTHMO a sepmHHH C BHCOTY CD. 3 npSIMOKYTHoro TpHKYTHHKa ACD, SIK~o KYT a. rocTpHii, ,u;icTaeMo:
CD= b sin a.
(Man. 265, |
a). HK~o KYT a. Tynu:ii, TO |
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CD = b sin (180° - a.) = |
b sin a. |
(Man. 265, |
6). AHanori11Ho a TpHKYTHHKa BCD AicTaeMo: |
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CD= a sin~· |
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c D |
A 8 |
c |
A 0 |
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t5) |
MaJI. 265
§ 12. Poaa'.11ayaaHH.H TpHKYTHHKiB |
175 |
0Tme, a sin~= b sin a. 3Bi,n;c:u |
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a |
sin fl |
--. |
sin a |
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AHaJiori'IHO,n;oBe,n;eMo piBHiCTh |
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c |
sin fl |
sin y |
,ll;JI.11 ,Zl;OBe,ZJ;eHH.11 Tpe6a rrpoBeCTH BHCOTY TPHKYTHHKa 3 BepIIIHHH A. TeopeMy ,n;oBe,n;eHo.
a)
Man. 266
3 a ,ZI; a 'Ia (13). ,lI;oBe,ZJ;iTh, w;o B TeopeMi CHHyciB KOa<He 3
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a |
b |
c |
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2R |
R |
- pa- |
TphOX Bl,ZJ;HOIIIeHh -.-, -.-A'-.- |
,ZJ;OplBHI06 |
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s1n a |
sin p |
sin y |
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,n;iyc KOJia, on:ucaHoro HaBKOJIO TPHKYTHHKa. |
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P o 3 B'.11 3 a H H .11. |
IIpoBe,n;eMo |
,n;iaMeTp |
BD (MaJI. |
266). |
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3a BJIRCTHBiCTIO KYTiB, BITHCaHHX y KOJIO, KYT rrp:u BepIIIHHi D rrp.11MOKYTHoro TPHKYTHHKa BCD ,n;opiBHI06 a6o a, .11Km;o
TO'IKHA i D JiemaTh 3 |
o,n;Horo 6oKy Bi,n; |
rrp.11Moi BC (MaJI. |
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266, a), aoo·-180° - |
a, .llKlll;O BOHH JiemaTh |
3 |
pi3HHX 6oKiB Bi,n; |
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rrp.11Moi BC |
(MaJI. |
266, |
6). Y rrepmoMy |
B:una,n;Ky BC = |
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= BD sin a, |
y ,n;pyroMy BC= ,BD sin (180° - |
a). |
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OcKiJihKH |
sin (180° - |
~sin a, TO ,n;JI.11 |
6y,n;L-.11Koro BH- |
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rra,n;Ky a = 2R sin a. To6To a
sin a
2R,
w;o ii TP~ 6yJio ,n;oBecT:u.
111. CIIIBBl,ll,HOIIIEHH.H MDK KYTAM:U TP:UKYTH:UKA
I IIPOT:UJIEmH:UM:U CTOPOHAM:U
Y rpu1Cyniu1Cy nporu 6iJ£bU£OZO Kyra J£e:HCUTb 6iJ£bU£a cropona, nporu 6iJ£bU£oi croponu J£e:HCUTb 6iJ£bU£uu KYT.
Hexali a i b - ,ZJ;Bi cTopoH:u TpHKYTHHKa i a, ~ - npoT:uJiemHi
9 KJiac |
176 |
'iMKYTH. }l;oBe,n;eMo, ~o KOJIH |
a > ~. TO a > b. I HaBnaKH, .HK~o |
a> b, TO a>~· |
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.HK~o KYTH a i ~ rocTpi (MaJI. 267, a), TO KOJIH a>~. 6y,n;e |
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sin a > sin ~· A ocKiJILKH |
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sin a |
sin~ |
- a - = - b - , |
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TO a> b. HK~o KYT a TynHH (o6H,n;Ba KYTH He MomyTL 6yTH TynHMH), TO KYT 180° - a rOCTPHH (MaJI. 267' 6). IlpHl.IOMY KYT 180° - - a 6iJibUIHH Bi,ZI; KyTa ~ .HK 30BHiUIHiH KYT TpHKYTHHKa, He cyMiJKHHH 3 KYTOM ~. 3Bi,Zl;CH sin a= sin (180° - a ) > sin~· I MH 3HOBY MaeMo, ~o a > b.
[ {
A |
B |
8 |
a) |
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tf} |
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MaJI. 267 |
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}l;oBe,n;eMo o6epHeHe TBep,n;meHHSI. Hexaif a > b. Tpe6a ,n;oBeCTH, |
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~o a > ~· IlpHilYCTHMO, ~o a ~ ~· .HK~O a = |
~. TO TPHKYTHHK |
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piBHo6e,n;peHHH i a = |
b. HK~o a < ~. TO aa |
,n;oBe,n;eHHM a < b. |
B ,n;Box BHna,n;Kax ,n;icTaJIH cynepet.1HiCTb, OCKiJibKH aa npHny~eH HSIM a> b, OTJKe a>~. ~o ii Tpe6a 6yJIO ,Zl;OBeCTH.
3 a ,n; a q a (17). }l;oBe,n;iTb, ~o KOJIH y TPHKYTHHKY e TYIlHH KyT, TO npOTHJiemHa ifoMy cTopoHa Haif6iJILma.
p 0 3 B'SI 3 a H H SI. y TPHKYTHHKY MOJKe 6yTH TiJibKH 0,Zl;HH TynHif KyT. 0Tme, BiH 6iJibIIIHii Bi,n; 6y,n;L-SIKoro a ,n;Box iHmHx KYTiB. A ~e oaHat.1ae, ~o npoTHJiemHa ifoMy cTopoHa 6iJibma Bi,n; 6y,n;b-SIKOl 3 ,Zl;BOX iHIIIHX CTopiH TpHKYTHHKa.
112. P03B'.H3YBAHH.HTPHKYTHHKIB
PooB'S1ayBaHHS1TPHKYTHHKiB noJIS1rae y aHaxo,n;meHHi HeBi,n;o- MHX CTOpiH i KyTiB TPHKYTHHKa aa Bi,Zl;OMHMH iforo KyTaMH i CTopoHaMH. IIoaHat.1aTHMeMo cTopoHu TpHKYTHHKa t.1epea a, b, c, a npoTHJieJKHi 'iMKYTH - qepea a, ~. y (MaJI. 268).
3 a A a q a (26). 1) y TPHKYTHHKY ,n;aHo CTOPOHY a= 5 i ,Zl;Ba KYTH: ~ = 30°' 'V = 45°. 3HaH,n;iTb TpeTiH KYT i iHmi ,Zl;Bi CTOpOHH.
§ 12. Poaa'naysaHHHTPHKYTHHKUJ |
177 |
Man. 268 |
Man. 269 |
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p 0 3 e'SI 3 a H HSI. |
OcKiJibKH cyMa KYTie |
TJ>HKYTHHKa |
µ;opiBHI06 180°' TO TpeTiH KYT a µ;opiBHI06: a = |
180° - ~ - |
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- y = 180° - 30° - |
45° = 105°. |
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3HaIO'IH CTOpOHY i eci TJ>H KYTH, aa TeOpeMOIO CHHyciB 3H8XO,D;HMO µ;ei iHmi CTOPOHH:
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b = |
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sin ~ |
= |
5 · |
sin 30° |
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~ 5 |
· |
0,500 |
~ 2,59; |
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a · - .- |
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0 , 966 |
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Sill ct |
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Sin 1050 |
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sin y |
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0,707 |
~ 3' 66 • |
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c = a · -.- |
~ 5 · o 966 |
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Sill ct |
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3 a A a 'Ia |
(27). |
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y |
TJ>HKYTHHKy |
lJ,8HO II.Bi CTOpOHH: |
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a = 12, b = 8 i |
KYT Mim HHMH y = 60°. 3HaHlJ,iTb iHmi lJ,Ba |
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KyTH i TpeTIO CTOpOHY. |
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Po a e's.a a H HSI. TpeTIO cTopoHy aHaXOlJ.HMO aa Teope- |
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MOIO KOCHHYCiB: |
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c = |
.ja2 + |
b2 - |
2ab cosy .=_:J144+ 64-2 · 12 · 8 · 0,500 = |
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= |
-v112 ~ 10,6. |
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Tenep, Ma1011u TPH cTopouu, aa TeopeMoIO Kocuuyci.B aHa- |
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XOAHMO KOCHHYCH II.BOX HeBiAOMHX KYTiB i caMi KYTH: |
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cos a = |
b 2 |
+ c 2 - |
a 2 |
~ 0,189. |
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.2bc |
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3BilJ,KH a= 79°, ~ = |
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180° - |
a - |
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y:::::::; |
41°. |
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3 a A a 11 a |
(28). |
5) |
Y |
TPHKYTHHKY |
lJ,aHo II.Bi |
cTopouu: |
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a = |
6, b = |
8 i npoTHJieJKHHH CTOpoui a KYT a = |
30°. 3uaii- |
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lJ,iTb iHmi lJ,Ba |
KYTH |
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TJ)eTIO |
CTOpouy. |
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p 0 3 e'SI 3 a H H B.. |
3a TeOpeMOIO CHHyciB 3HaXO,D;HMO 3Ha- |
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'leHHSIsin ~ : |
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sin~= : • sin a= : . sin 30°:::::::; 0,667.
D;boMy 3Ha11eHHIO CHHyca BilJ,noeiAaIOTb lJ,Ba KYTH: ~I :::::::; 42 ° i
~2 ~ 138°.
PoarJIS1HeMo cno'laTKYKYT ~1 :::::::; 42°. 3a HHM auaxolJ,HMo TpeTiH KyT y1 = 180° - a - ~1 = 108° i aa TeopeMoIO cHHyciB TpeTIO cTopoHy
= |
a sin y |
1 |
= |
6 |
. sin 108° |
_, 6 . 0 ,951 _, |
ll 4 |
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Ci |
sin ct |
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sin 30° |
"' |
0,500"' |
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AHaJiori'IHO, aa KYTOM |
~2 = 138°, |
aHaXOAHMO |
Y2 = 12° i |
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C2 :::::::; 2,49. |
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7 r eoMeTpiJI , 7. 9 K JI .
9 KJiac |
178 |
3 a y B a m e H H .H. |
Mu 6a11HMO, w;o ~.H aa,Z1;a11a Ha Bi,D;MiHy BiA |
nonepe,D;HiX Mae ,Zl;Ba p03B'.H3KH(MaJI. 269). IlpH iHIIIHX 11HCJIOBHX 3Ha11eHH.HX, HanpHKJia,D;, KOJIH CG ;::: 90°' aa,Z1;a11a MO>Ke MaTH JIHIIIe O,Zl;HH poaB'.H30Ka6o 30BCiM He MaTH p03B'.HaKiB.
3 a A a 11 a |
(29). |
1) J!a~o :PH CTOpOHH TpHKYTHHKa: |
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a = |
2, b = |
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c = |
4. |
3HaH,D;1Tb uoro KYTH. |
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p 0 3 B'SI 3 a H HSI. |
KyTH 3HaXO,ll;HMO aa TeopeMOIO KOCH- |
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HyCiB: |
b2 + c2 - ·aZ |
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0,875, 3Bl,Zl;KH CG~ 29 °. |
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Aaanori'•rnoaHaXO,Zl;HMO cos ~ = |
0,688. 3Bi,D;KH ~ ~ 47° i |
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v ___:. |
180° - |
47° - |
29° = 104°. |
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?KOHTPO~bHI3AITHTAHHH
1.,lJ;oBe,D;iTb TeopeMy KOCHHyciB.
2.,lJ;oBe,D;iTb, w;o KBa,D;paT CTOPOHH TPHKYTHHKa ,Zl;OpiBHI06 cyMi KBR,Zl;paTiB ,Zl;BOX iHIIIHX CTopiH • +. Il0,Zl;B06HHH ,D;o6yTOK O,Zl;Hie'i a ~ux cTopiH Ha npoeK~iIO ,D;pyro'i.BiA 1wro aaJie>KHTL 3HaK
•+• a6o •-•?
3.,lJ;oBe,D;iTb TeopeMy CHHyciB.
4.,lJ;oBe,D;iTL, w;o B ,Zl;OBiJILHOMY TPHKYTHHKY npoTH 6iJI1>IIIo'i CTOpOHH Jie>KHTb 6iJILIIIHH KYT i npOTH 6iJil>IIIOro KyTa .iJ:emHTb
.:iJILIIIa CTOpOHa. |
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y |
3AAAqI |
1.CropoHH TPHKYTHHKa 5 M, 6 M, 7 M. 3aaii,D;iTL KOCHHYCH KyTiB TpHKYTHHKa.
2.Y TPHKYTHHKY ,D;Bi cTopoHH ,Z1;opiBH1010T1> 5 M i 6 M, a cuHyc KyTa Mim HHMH ,Zl;OpiBH106 0,6. 3aaii,D;iTb TpeTIO CTopoay.
3.CTopoHH TpHKYTHHKa ,D;opiBHIOIOTL a, b, c. ,lJ;oBe,D;iTL, w;o KOJIH
+b2 > c2, To KyT, npoTHJiemau:H cTopoHi c, rocTpHii. Konu
+b2 < c2, TO KyT, npOTHJie>KHHH CTOpOHi c, TYilHH.
4.,lJ;aHo ,D;iaroaaJii napaJieJiorpaMa c i d i KYT Mim HHMH a . 3HaiiAiTL CTOPOHH napaJieJiorpaMa.
5. ,lJ;aHO CTOPOHH napanenorpaMa a i b i O,ll;HH 3 KyTiB CG. 3Haii- ,D;iTb ,D;iaroHaJii napaJienorpaMa.
6.CropoHH TPHKYTHHKa 4 M, 5 M i 6 M. 3Haii,D;iTL npoeK~i'i cTopiH 4 M i 5 M Ha np.HMy, Ha SIKi:H Jie>KHTb CTOpoHa 6 M.
7.,D;aao CTOPOHH TPHKYTHHKa a, b, c. 3aaii,D;iTb BHCOTY TPHKYTHHKa, onyw;eay Ha cTopoHy c.
8.3aaii,D;iTb BHCOTH TpHKYTHHKa B aa,Z1;a11i 1.
9.3HaH,D;iTb Me,D;iaHH TpHKYTHHKa B aa,D;a'li1.
10!' 3HaH,D;iTb 6iceKTpHCH TpHKYTHHKa B 3a,D;a'li1.
11!' HK aMiHI06TLCSI cTopoHa AB TpHKyTH~Ka ABC, SIKw;o KYT C apocTae, a ,Zl;OB>KHHH cTopiH AC i BC aanumaIOTLCSI 6ea aMiH (MaJI. 270)?
§ 12, PoaB'11ayBaHH11TpHKYTHHKiB |
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179 |
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12. Y TpHKYTHHKY |
ABC: AB= 15 CM, |
AC= 10 CM. qH Mome |
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sin f) = |
! ? |
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13. ,JJ,oBe,ll;iTb, ~o B TeopeMi |
CHHyciB KO)l(He 3 TpbOX BiAHOWeHb |
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a |
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2R |
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R |
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-.- , ---11.• -.- |
,11;op1BHIOe |
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pa,11;1yc KOJia, onHcaHoro |
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sm ct sm., |
sm y |
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HaBKOJIO TPHKYTHHKa.
14.HK aHaiiTH pa,11;iyc KOJia, onucaHoro HaBKOJIO TPHKYTHHKa, 3HaIO'IHiioro CTOPOHH? 3Haii,11;iTb pa,11;iyc KOJia, OilHCaHoro HaBKOJIO TpHKYTHHKa ia CTOpoHaMH 5 M, 6 M, 7 M.
15.IloHCHiTb, HK aHaiiTH Bi,ll;cTaHb si,11; TO'IKHA ,11;0 He,11;ocTynHoi TO'IKHB (MaJI. 271), aHaIO'IHBi,11;cTaHb AC i KYTH a., fl.
16.IloHCHiTb, HK 3HaHTH BHCOTY x 6y,11;iBJii (MaJI. 272) aa KyTaMH a i ~ i Bi,ll;CTaHHIO a.
17.,JJ,ose,11;iTb, ~o KOJIH B TPHKYTHHKY e TYilHH KyT, TO npOTHJiemHa H:oMy cTopoaa Haii6iJihwa.
18. |
y |
TPHKYTHHKY |
ABC: LA= 40°; LB= 60°; |
L c = 80°. |
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HKa ia cTopiH |
TpHKYTHHKa HaH:6iJibWa, |
HKa - |
Haii:MeHwa? |
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19. |
y |
TPHKYTHHKY |
ABC |
CTOPOHH AB = |
5,1 M; |
BC = |
6,2 M; |
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AC = 7,3 M. HKHH a |
KyTiB TPHKYTHHKa |
aaii6iJibWHii, |
HKHii |
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HaiiMeHwHii? |
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20. |
IU;o 6iJibwe: OCHOBa 'IH6i'IHaCTOpoHa piBH06e,11;peHoro TPHKYT- |
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HHKa, HK~O npHJierJIHH ,11;0 OCHOBH |
KYT 6iJibWHH Bi,11; 60°? |
21. |
Y TPHKYTHHKY ABC KYT C TynHii. ,ll;oBe,11;iTb, ~o KOJIH TO'IKaX |
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nemHTb Ha cTopoHi AC, TO BX< AB. |
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22. |
Y TPHKYTHHKY ABC KYT C Tynuii. ,ll;ose,11;iTb, ~o KOJIH TO'IKaX |
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nemHTb Ha cropoHi AC, a TO'IKaY - |
Ha cTopoai BC, TO XY < |
<AB.
23. Ha cTopoHi AB TPHKYTHHKa ABC noaHa'leHoTO'IKYD. ,ll;oseAiTb, ~o BiApiaoK CD MeHwHii xo'la6 aa o,11;ay ia cTopiH: AC a6o BC.
24tr ,ll;aHo TpHKYTHHK ABC. CD - |
Me,11;iaHa; np0Be,11;eaa AO cropoHH |
AB. ,ll;oBeAiTb, ~o KOJIH AC > BC, To KYT ACD MeHWHH Bi,11; |
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KyTa BCD. |
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A |
8 |
Mu. 270 |
Man. 271 |