Геометрия7_9
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160 |
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MaJI. |
239 |
p 0 3 B'Si 3 a H H Si (MaJI. |
239). y TPHKYTHHKiB ABC i |
A1B 1C1 KYT npH BepmHHi C cniJibHHH, a KYT CA,B, .a;opiBHI06 KYTY CAB, HK Bi.a;noBi.a;Hi KYTH npu napaneJibHHX AB i A,B, i ci•rni:H AC. OT:me, !:::.. ABC= !:::.. A,B,C aa .a;BoMa KyTaMH.
104. 03HAKA IlOAIBHOCTI TPHKYTHHKIB
3A ABOMA CTOPOHAMH I KYTOM MDK HHMH
T e o p e M a 11.3. HK~o oai cToponu oonozo rpuKyTH.UJca nponop1fi'iini 080M CTOpOH.aM opyzozo TpU1'YT1tU1'a i "YTU, YTBopeni lfUMU CTOpOKaMU, piBH.i, TO TpU1'YTKU1'U nooi6ni.
,lJ; 0 Be ,!I; e H H Si (aHaJiori'IHe ,!l;O ,!l;OBe,D;eHHH TeopeMH 11.2). Hexaii y Tp1rnyTHHKiB ABC i A1B1C1: LC= LC, i AC= kA,C,, BC= kB,c,. ,lJ;oBe.a;eMO, w;o !:::.. ABC CXJ !:::.. A,B,C,.
3acTocyeMo .a;o TPHKYTHHKa A,B,C, nepeTBopeHHH no.a;i6aocTi 3 Koe<l>i~i6HTOM no.a;i6HOCTi k, HanpHKJia,D; roMOTeTiIO (MaJI. 240). IlpH ~bOMY .a;icTaHeMo .a;eHKHH TPHKYTHHK A 2B2C2, m;o .a;opiBHI06 TpHKYTHHKY ABC. CnpaB.a;i, ocKiJibKH nepeTBopeHHH no.a;i6HocTi
a6epirae KYTH, |
TO L C2 = |
LC,. OT:me, y |
TpHKYTHHKiB ABC i |
A 2B2C2 LC= L |
C2. ,lJ;aJii, |
A2C2 = kA1C1 = |
AC, B2C2 = kB1C1 = |
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MaJI. 240
§ 11. Ilo)li6HiCTb cPiryp |
161 |
[
Man. 241
=BC. 3BiACH BHilJIHBae, m;o TPHKYTHHKH ABC i A2B£2 piBHi aa nepmoIO 03HRKOIO (aa ABOMa CTOpOHRMH i KYTOM Mim HHMH).
OcKiJI:LKH TPHKYTHHKH A1B1C1 i A2B2C2 roMOTeTH'IHii, OTme, noAi6Hi., a TJ>HKYTHHKH A:i82C2 i ABC piBHi i TOMY Tem noAi6Hi, To TPHKYTHHKH A1B1C1 i ABC noAi6Hi. TeopeMy AOBeAeHo.
0 |
3 a A a q a (31). Y TJ>HKYTHHKY ABC a rocTpHM KYTOM C |
npoBeAeHi BHCOTH AE i BD (Man. 241). ,ll;ose,u;iT:L, m;o .!:::. ABC= |
|
= !:::.EDC. |
Po as'a a a H Ha. Y TpHKYTHHKiB ABC i EDC KYT npu BepmHHi c cniJI:LHHH. ,ll;oBeAeMO nponop~HiCT& CTOpiH TpHKYTHHKiB, sKi npunaraIOT& AO ~oro KyTa. MaeMo EC = = AC cosy, DC= BC cosy. To6To cTopouu, m;o npHJisraIOT& AO KyTa C, y TPHKYTHHKiB nponop~ui. 0Tme, .!:::. ABC = = .!:::. EDC aa ABOMa cropouaMH i KYTOM Mim HHMH.
105. 03HAKA IIOfUBHOCTI TPHKYTHHKIB
3A TPhOMA CTOPOHAMH
Teo p e Ma 11.4. H'1U40 CTOpoHu oiJHozo rpu.KyTHuxa nponopJfiuHi cropoHaM iJpyzozo, ro rad rpuKyrHU'ICUnoiJi6Hi.
,n; o Be A e ~Ha (aHanori'IHe AO AOBe,AeHHa reopeMH 11.2).
Hexaii y TpHKYTHHKiB ABC i A1B1C1: AB= kA1Bi, AC= kA1Ci, BC= kB1C1. ,ll;o:eeAeMo, m;o .!:::. ABC= .!:::. A1B1C1.
3aCTocyeMO nepeTBopeHHS IlO,Ai6HOCTi 3 Koeclii~6HTOM UOAi6HOCTi k AO .!:::. A1B1Ci. HanpuKJiaA rOMOTeTiIO (MaJI. IlpH ~oMy AiCTaHeMo AeHKHii TPHKYTHHK A:iB.£2, m;o AOpiBHIOe TpHKYTHHKY ABC. CnpaBAi, y TPHKYTHHKiB Bi,u;nosi,u;ui cropoHH piBHi:
A2B2 = |
kA1B1 = AB, A2C2 = kA1C1 =AC, B.£2 = kB1C1 = BC. |
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3Bi,u;CH |
BHilJIHBae, m;o TPHKYTHHKH |
piBHi aa TpeT&OIO 03HRKOIO |
(aa Tp&oMa CTOpOHRMH). |
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OcKiJI&KH TpHKYTHHKH A1B1C1 i |
A2B2C2 roMOTeTH'IHi,i OTme, |
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noAi6Hi., a TPHKYTHHKH A:iB.£2 i ABC pisui i TOMY rem noAi6ui, TO TPHKYTHHKH A1B1C1 i ABC noAi6ui. TeopeMy AOBeAeuo.
6 feoMeTpbt, 7-9 KJI.
9 KJI8C |
162 |
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0
MaJI. 242
3 a A a 11 a (36). ,IJ;oBeAiTI>, m;o y IIOAi6HHX TPHKYTHHKiB rrepHMeTpH Bi.AHOCSIT:&CR, $1K BiAIIOBi.AHi CTOpoHH.
Po as'a a a H Ha. Hexaii ABC i A 1B1C1 - IIOAi6Hi TPH· KYTHHKH. ToAi CTopoHH TpHKYTHHKa A1B1C1 rrporrop~iiHi
cropoHaM |
TpHKYTHHKa |
ABC, T06To A1B1 = kAB, B1C1 = |
= kBC, |
A1C1 = kAC. |
,lJ;oAaBmH ~ pisuocTi 11011JieHHo, |
AicTaHeMo:
A1B1 + B1C1 + A1C1 ==. k (AB+ BC+ AC).
3Bi.ACH
A,B, + B,c, + A,C, |
A,B , |
A,c, |
B1C1 |
AB+ BC + AC |
= AB = |
AC = |
BC ' |
To6To rrepHMeTPH TpHKyTHHKiB BiAHOCS1T1>ca, RK BiAIIOBiAHi CTOpOHH.
106. IlOJJ;IBHICTb IlP.HMOKYTHHX TPHKYTHHKIB
Y rrpRMoKyTHoro TPHKYTHHKa OAHH KYT rrpaMHH. ToMy, aa TeopeMOIO 11.2, iJ1ui noiJi6HOCri iJsox npl&1l&01C1fl'HUX TpU1C1fl'1'U1Ci6 iJocuT.,,, UfOO y """ 6yJJo no pUJHOMY ZOCTpoM.y rcyry.
3a AOnoMoroIO ~e'i 03H&KH rroAi6uocTi rrpRMoKyTHHX TPHKYTHHKiB AOBeAeMo AeRKi cniBBi.Auomeuua y TPHKYTHHKaX.
Hexaii ABC - npgMoKyTuHii TPHKYTHHK a rrpaMHM KYTOM C. IlposeAeMo sucoTy CD a sepmHHH rrpaMoro KyTa (MaJI. 243).
TpHKYTHHKH ABC i CBD MaIOTI> cniJii.HHii KYT rrpH sepmuHi B. Orme, oouu IIOAi6Hi: /:::,.ABC= /:::,. CBD. 3 rroAi6HocTi TpHKYTHHKiB BHIIJIHB86 Bi.AIIOBi.AHHX CTOpiH:
a6o BC = ./AB ·BD.
~e crriBBi.AHomeHH$1 cl>opMyJIIOGTl>C$1 T8K: JCQ,TeT npl&1l&01CYT1'0ZO rpu1CYT1'U1CQ, e ceJH!iJHi111. nponop1fiU1'UM 111.i3rc zinoreKyao10 i npoe1C1fie10 lf"Ozo 1C4Tera Ka zinoreKyay.
IlpaMOKYTHi TPHKYTHHKH ACD i CBD Tem IIOAi6ui: y HHX
§ 11. lfo'Ai6HiCTb ciJiryp |
163 |
MaJI. 243 MaJI. 244
piBHi rocTpi KYTH npu BepwHHax Ai C. 3 no,D;i6Hocri ~Hx TpHKYTHHKiB BHilJIHBae nponop~iHHiCTh IX CTOpiH:
AD |
CD |
CD = |
_/ |
CD = |
BD' a6o |
yAD ·BD. |
u;e cniBBi,Zl;HOWeHHJI cPOPMYJIIOIOTh TaK: BU.Cora npR.MOICflTHOZO TpU1Cf1THU1'a, npoBeiJeHa 3 BepraUHU npR.MOZO xyra, e cepeiJHiM nponopifiUHUM Mi31t npoexifiR.Mu xareriB Ha zinoreHyay.
,lJ;oBe,D;eMo TaKy BJiaCTHBiCTh 6iceKTpHcH TPHKYTHHKa: 6icexrpuca rpuKyTHuxa iJiJiur-,, nporuJt.e:llCHfl cropoHy Ha BiiJpiaxu, nponoplfiUHi iJBOM iH1"UM CTOpoHaM.
Hexaii CD - 6iceKTpuca TPHKYTHHKa ABC (Man. 244). .HK~o TPHKYTHHK ABC piBH06e,D;peHHH 3 OCHOBOIO AB, TO ccl>opMym.osaHa BJIRCTHBiCTh 6iceKTPHCH O'leBH)l;Ha, OCKiJibKH y ~hOMY BHDRAKY 6iceKTpuca CD s i Me,D;iaHoIO.
Po3rJIJIHeMo aaraJILHHH BHDRAOK, KOJIH AC =I= BC. OnycTHMo 3 BepWHH A i B nepneHAHKYJIRPH AF i BE Ha npsMy CD.
IlpsMoKyTHi TPHKYTHHKH ACF i BCE noAi6Hi, ocKiJihKH y HHX piBHi rocTpi KYTH npH sepWHHi c. 3 no,D;i6HOCTi TpHKyTHHKiB BHDJIHBae nponop~HHiCTh CTOpiH:
AC AF
BC= BE'
IlpsMOKYTHi TpHKYTHHKH ADF i BDE TaKo:m no,D;i6Hi: y HHX KYTH npu sepmHHi D pisHi RK eepTHKaJILHi. 3 no,D;i6HocTi TpHKYTHHKiB BHnJIHBae nponop~iiHicTh cropiH:
AF AD
BE= BD'
IlopiBHJIBWH ~ pieHiCTh 3 nonepe,D;HhOIO, AiCTaHeMo:
AC AD |
AC BC |
BC= BD a 60 |
AD= BD' |
To6To BiAPi3KH AD i BD nponop~Hi cTopoHaM AC i BC, ~o ii TPe6a 6yJIO AOBecTH.
9 KJiac |
164 |
107. KYTH, BIIHCAHI B KOJIO
KyT poa6HBae IIJIOIIJ;HHY Ha ,o;Bi 11acTHHH. Ko~ma a ~x 11acTHH Ha3HBaeT&ca nJtoCH:UM KYTOM. Ha MaJIIOHKy 245 aamTpHXOBaHo O.D;HH a IIJIOCKHX KyTiB i3 cTopoHaMH a i b. IIJiocKi KYTH i3 c11iJibHHMH
CTOpOHaMH Ha3HBaIOTbC8 aon061Ul.JlbltUMU.
HKw;o IIJIOCKHH KyT e 11acTHHOIO 11iBIIJIOIIJ;HHH, TO iioro rpa.o;yc-
HOIO MipoIO Ha3HBa6TbC8 rpa.o;ycHa Mipa 3BH'laiiHoroKyTa 3 |
THMH |
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CaMHMH CTOpOHaMH. BBalKaIOTb, IIJ;O KOJIH IIJIOCKHH KYT MiCTHTb |
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IIiBIIJIOIIJ;HHy, |
TO iioro rpa.o;ycHa Mipa .o;opiBHI06 360° - |
a, .o;e |
a - rpa,u;ycHa |
Mipa .D;OIIOBH8JlbHOro IIJIOCKOro KyTa (MaJI. |
246). |
b
MaJI. 245 |
Mu. 246 |
A
MaJI. 247 MaJI. 248
~enrpaJtbltUM H:Y'l'OM y KOJii Ha3HBa6TbC8 IIJIOCKHH KYT 3 BepmH-
HOIO y iioro ~eHTpi. qaCTHHa KOJia, pooMiw;eHa BCepe.o;HHi IIJIOCKOrO KyTa, Ha3HBa6TbC$1 ayz010 H:OJta, UJ;O Bi.D;IIOBi,o;ae ~bOMY ~eHTpaJib HOMy KYTY (MaJI. I'paayc1t010 Mip010 ayzu KOJia Ha3HBa6TbC8 rpa.o;ycHa Mipa Bi.o;IIOBi.D;HOro ~eHTpaJibHOro KyTa.
KyT, BepmHHa sKoro JielKHTb Ha KOJii, a cTopoHH 11epeTHHaIOTb ~e KOJio, Ha3HBaeT&cs BnucanuM y H:oJto (MaJI. KyT
§ 11. IIo,iti6H.icTb <l>iryp |
165 |
BAC Ha MaJIIOHKY 248 BnHcaHo B KOJIO. Horo uepmHHa A ;xemHTb Ha KOJli, a CTOpoHH nepeTHHaIOTb KOJIO B TO'IKaXB i c. KamyTb TaKom, m;o KYT A cnHpaeTbCa: Ha xopp;y BC. Ilpa:Ma BC poa6HBae KOJIO Ha ,D;Bi p;ym. D;eHTpaJibHHH KYT, m;o BiAnOBi,D;ae Tiii 3 IUIX p;yr, a:Ka He MiCTHTb TO'IKYA, HaaHBa6TbCa: u;eHTpaJXbHHM KYTOM, .R1'UU eianoewae iJanoMy enucanoMy 1'yry.
Teo p e Ma 11.5. Kyr, anu.caHu.ri y xo;io, iJopieHJOe noiioeu.Hi eiiJno6iiJHOZO qeHTpaJibHOZO 'ICyra.
Ao Be A e H H s. PoarJXa:HeMo cnot1aTKY oKpeMHH BHnap;oK, KOJIH op;Ha ia CTopiH KyTa npoXO,Zl;HTb qepea u;eHTP KOJla (MaJl. 249, a). TpHKYTHHK AOB piBHo6e,D;peHHii, 6o B Hboro cropoHH OA i OB piBHi $IK pap;iycH. A TOMY KYTH A i B TPHKYTHHKa piBHi. A qepea Te m;o 'ixcyMa p;opiBHIOe aoBHimHbOMY KYTY TPHKYTHHKa npH BepmHHi 0, TO KYT B TPHKYTHHKa p;opiBHI06 noJXOBHHi KyTa AOC, m;o ii Tpe6a 6yno p;oBecTH.
B
0
|
0 |
8} |
a) |
6) |
MaJI. 249
3araJibHHH BHila,D;OK 3BOp;HTbCJI AO poarJia:HyToro OKpeMoro, HKm;o npo:eecTH p;onoMimmm p;iaMeTp BD (MaJI. 249, 6, e).
)J;JIH BHnap;Ky, nop;aHoro Ha MaJIIOHKY 249, 6, Ma6MO:
L ABC = L CBD + L ABD =
= TL COD + TL AOD =
1
= 2 LAOC.
)J;JIH BHDa,D;KY'nop;aHoro Ha MaJIIOHKY
249, 8, Ma6MO:
L ABC = |
L CBD - L ABD = |
= -12 L COD - -12 L AOD = |
|
= |
1 LAOC. |
|
2 |
TeopeMy p;oBep;eHo noBHiCTIO. |
MaJI. 250 |
9 KJiac |
166 |
3 TOOpeMH 11.5 BHilJIHB86, ~o 6nUCaHi ICYTU, CTOpDHU IUCUX npoX001'Tb 1f.epea T01f.1CU A i B 1e011.a, a BeprauHu Jie31taTb a oiJHOzo 601cy BiiJ npsiMo'i AB, pi6Hi (MaJI. 250).
3oKpeMa, 1CyTu, UfO cnupalOTbC1' Ha iJiaMeTp, npsiMi.
108. IIPOIIOPD;IHHICTb BIAJ>I3KIB XOP,l.I;
I CI'IHHX KOJIA
H1CUfO xopiJu AB i CD ICOJia nepeTUHalOTbCsi |
6 T01f.1fi S, To |
AS • BS = CS ·DS. |
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Cno'laTKY p,oBe,u,eMo, ~o TpHKYTHHKH ASD i CSB no,u,i6Hi |
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(MaJI. 251). BnucaHi KYTH DCB i DAB pisHi aa |
HacJii,u,KOM ia |
TeopeMH 11.5. KyTH ASD i BSC piBHi .RK BepTHKaJibHi. 3 piBHOCTi HaaBaHHX KyTiB sunJiusas, ~o TpHKYTHHKH ASD i CSB no.id6Hi.
MaJI. 251 MaJI. 252
3 no,u,i6HOCTi TPHKYTHHKiB BHilJIHBa6 nponop~i.R
DS AS
BS= CS.
AS ·BS = CS · DS,
~o H: Tpe6a 6yJIO p,oBecTH.
HICUfO a T01f.1CU P iJo 1e011.a npoBeiJeHo iJBi ci1f.Hi, UfO nepeTuHaIOTb 1e011,o BiiJnoBiiJHo 6 T01'1CaX A, B i C, D, To
AP · BP= CP · DP.
HexaH: TO'IKHA i C Haii6JIHm'lip,o TO'IKHP TO'IKHnepeTHHY ci'IHHX a KOJIOM (MaJI. 252). TpHKyTHHKH PAD i PCB no,u,i6Hi. Y HHX KYT npH BepmHHi P cniJihHHii, a KYTH npH sepmHHax B i D piBHi aa BJiaCTHBOCT.RMH KyTiB, BnHcaHHX y KOJIO. 3 no,u,i6HOCTi TPHKYTHHKiB BHilJIHBa6 nponopaj.R:
PA PD
PC= PB"
3Bip,cH PA ·PB= PC· PD, ~o H: Tpe6a 6yJio ,u,oseCTH.
§ 11. |
IIo,n;i6HicTb ci>iryp |
167 |
? |
KOHTPO~bHI3AilHTAHHH |
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1.ru;o TaKe nepeTBOpeHH$1 IlO,D;i6HOCTi?
2.ru;o TaKe roMOTeTia (:a;eHTP roMOTeTii, Koecl>i:a;ieHT roMOTeTiI)?
3. .D;oBe,D;iTL, :r:a;o roMOTeTia e nepeTBopeHH8M IlO,D;i6HOCTi.
4.HKi BJiacTHBOCTi nepeTBopeHHa no,D;i6HocTi BH aHaeTe? .D;oBe- ,D;iTL, :r:a;o nepeTBopeHHa IlO,D;i6HOCTi a6epirae KYTH Mim niBnpaMHMH.
5.HKi cl>irypH Ha3HBaIOTLca no,D;i6HHMH?
6. .HKHM 3HaKOM Il03Ha'la6TLC$1IlO,D;i6HiCTL <l>iryp? .HK aanHcyeTLCa IlO,D;i6HiCTL TpHKYTHHKiB?
7.Ccl>opMyJIIOHTe i ,D;OBe,D;iTL oaHaKy no,D;i6HocTi TPHKYTHHKiB aa ,Zl;BOMa :KyTaMH.
8.Ccl>opMyJIIOHTe i ,D;OBe,D;iTL oaHaKy no,D;i6HocTi TPHKYTHHKiR aa ,Zl;BOMa CTOpOHaMH i KYTOM Mim HHMH.
9.Ccl>opMyJIIOHTe i ,D;OBe,D;iTL oaHaKy no,D;i6HocTi TPHKYTHHKiB aa TPLOMa CTopoHaMH.
10. .D;oBe,D;iTL, :r:a;o KaTeT npaMOKYTHOro TPHKYTHHKa e cepe,D;H6 nponop:a;i:HHe Mim rinoTeHyaoro i npoeK:a;iero :a;Loro KaTeTa Ha rinoTeHyay.
11. .D;oBe,D;iTL, :r:a;o BHCOTa npaMOKYTHOro TpHKYTHHKa, npoBe,u;eHa a BepmHHH npaMoro KyTa, e cepe,D;He nponop:a;i:HHe Mim npoeK- :u;iaMH KaTeTiB Ha rinoTeHyay.
12. .D;oBe,D;iTL, :r:a;o 6iceKTpHca TPHKYTHHKa ,D;iJIHTL npoTHJie)lrny CTOPOHY Ha Bi,D;piaKH, nponop:a;iHHi ,Zl;BOM iHlliHM CTOpOHaM.
13.ru;o TaKe IlJIOCKHH KyT?
14.ru;o TaKe :a;eHTpaJILHHH KyT?
15. .HKHH KYT Ha3HBa6TLC$1 BilHCaHHM y KOJIO?
16. .D;oBe,D;iTL, :r:a;o BilHCaHHH y KOJIO KYT ,D;OpiBHI06 IlOJIOBHHi Bi,D;IlOBi,D;HOrO :a;eHTpaJILHOro KyTa.
17. .D;oBe,D;iTL BJiaCTHBiCTL Bi,D;piaKiB xop,D;, :r:a;o nepeTHHaIOTLCH, i
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BJiaCTHBOCTi Ci'IHHXBi,D;piaKiB. |
• |
3A.D;Aql |
1. Y peayJILTaTi roMoTeTii TO'IKaX nepexo,D;HTL y TO'IKYX', a TO'IKa Y - y TO'IKY Y'. HK aHaHTH :a;eHTp roMoTeTii, aK:r:a;o TO'IKHX, X', Y, Y' He JiemaTL Ha O,D;Hi:H npaMiH?
2. Y peayJILTaTi roMoTeTii TO'IKaX nepexo,D;HTL y TO'IKYX'. Ilo6y,D;y:HTe :a;eHTp roMOTeTii, 8KI:a;O Koecl>i:a;ieHT rOMOTeTii ,D;OpiBHI06 2.
3.HaKpecJiiTL TPHKYTHHK. IIo6y,D;yHTe roMoTeTH'IHHHHOMY TpHKYTHHK, IlpHHH8BlliH aa :a;eHTP roMOTeTii O,Zl;HY 3 :Horo BepmHH, a Koe<l>i:u;ieHT rOMOTeTii ,D;OpiBHI06 2.
4.Ha MaJIIOHKY 236 ao6pameHo nJiaH ca,D;H6H y MacmTa6i 1: 1000. BHaHa'ITepoaMipH ca,D;H6H (,D;OBJKHHY i mHpHHy).
9 KJiac |
168 |
5.IIJ;o HBJIH6 co6o10 cl>irypa, IlOAi6Ha TPHKYTHHKy?
6.y IlOAi6HHX TpHKYTHHKiB ABC i AiBiCi: LA= 30°, AB=
= 1 M, BC = 2 M, B iC i = 3 M. qOMY ,D;OpiBHIOIOTL KYT Ai i CTOpoHa ABi?
7. ,ll;oBeAiTL, m;o cl>irypa, no,D;i6Ha KOJiy, e KOJIO.
8~ ,ll;aHo KYT i scepe,D;HHi HLoro TO'IKYA. Ilo6YAYHTe KOJio, m;o AOTHKaeTLCH AO cTopiH KyTa i npoxo,D;HTL qepea TO'IKYA.
9~ BnHmiTL y AaHHii TpHKYTHHK KBaApaT, ABi sepIIIHHH S:Koro Jie»eaTL Ha O,Zl;Hiii CTOpoHi, a ABi iHIIIi sepIIIHHH - Ha ,Zl;BOX iHIIIHX CTOpoHax.
10.,ll;oBe,D;iTL IlOAi6HiCTL piBH06eApeHHX TPHKYTffHKiB, HKi MSIOTL pisHi KYTH npH sepIIIHHax, npoTHJie)KHHX OCHOBSM.
11.y ABOX piBHo6e,D;peHHX TPHKYTHHKiB KYTH Mim 6i'IHHMH cTopoHaMH piBHi. Bi'IHacTopoHa i ocHosa O,D;Horo TPHKYTHHKa Bi,Zl;IlOBi)l;HO ,Zl;OpiBHIOIOTL 17 CM i 10 CM; OCHOBa APYroro p;opiBHI06 8 CM. 3Haii,Zl;iTL iioro 6i'IHYCTOpoHy.
12.y TPHKYTHHKiB ABC i AiBiCi: LA= L A1, LB= L Bi,
|
AB= 5 M, BC= 7 M, A1Bi = 10 M, |
A1Ci = 8 M. |
3HaiiAiTL |
|
peIIITY CTOpiH TpHKYTHHKiB. |
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13. |
Poas's:miTL aa,D;a'ly 12 aa yMOBH, m;o AB= 16 CM, BC= |
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14. |
= 20 CM, AiB1 = 12 CM, AC -AiCi = |
6 CM. |
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,ll;oBe,D;iTL, m;o BHCOTa nps:MOKYTHOro |
TPHKYTHHKa, |
onym;eHa |
|
a sepIIIHHH nps:Moro KYTa, poa6Hsae :Horo Ha ABa TPHKYTHHKH, IlOAi6Hi ,D;aHOMy.
15.Ilps:Ma, napaJieJILHa cTopoHi AB TPHKYTHHKa ABC, nepeTHHae
iioro cTopoHy AC y To'laj.Ai. a CTopoHy BC y TO'laj.Bi. ,ll;oseAiTL, m;o 6. ABC = ,0,. AiBiC.
16.y TPHKYTHHK 3 OCHOBOIO a i BHCOTOIO h BilHCaHO KBa,D;paT TaK, m;o ABi :Horo sepIIIHHH JiemaTL Ha OCHOBi TpHKYTHHKa, a iHIIIi
ABi - Ha 6i'IHHX CTOpoHaX (MaJI. 253). 06'1HCIDTL CTOPOHY KBa,D;paTa.
17.Ilps:Ma, napaneJILHa cTopoHi AB TpHKYTHHKa ABC, AiJIHTL iioro cTopoHy AC y BiAHOIIIeHHi m : n, no'IHHaIO'IHsiA sepIIIHHH C. B s:KoMy BiAHomeHHi soHa AiJIHTL cTopoHy BC?
18.Y TPHKYTHHKY ABC npose,D;eHo Bi,D;piaoK DE, napaJieJILHHH cTopoai AC (KiHe~L D Bi,D;piaKa ne»eHTL Ha cTopoai AB, a
l?<i
A D
Man. 253 |
Man. 254 |
§ 11. IIoAi6aicTL <Piryp |
169 |
E - Ha cTopoHi BC). 3ttaif:AiTh AD, .11Km;o AB= 16 CM, AC=
=20 CM i DE = 15 CM.
19.Y aaAa'li18 attaif:AiTh BiAHOIIIeHH.11 AD : BD, .llKIIJ;O AC : DE= =55: 28.
20.3ttaif:AiTh AOBatHHY BiApiaKa DE y aaAa'li18, .11Km;o: 1) AC= =20cM,AB= 17cMiBD= ll,9cM;2) AC= lBAM,AB=
=15 AM i AD = 10 AM.
21.,D;iaroHaJii Tparre~il ABCD rrepeTHHaIOTbC.11 B TO'l~i E (MaJI. 254). ,lJ;oBeAiTh rroAi6HiCTb TPHKYTHHKiB BCE i DAE.
22.3HaHAiTb BiAHOIIIeHH.11 BiApiaKiB AiaroHaJii Tparreaji, Ha .llKi Botta poa6HBa6TbC.ll APYroro AiaroHaJIJIIO, .llKIIJ;O OCHOBH Tparre- ~il BiAHOC.llTbC.11 .llK m : n.
23.1Ip.11Ma, m;o rrpOXOAHTb qepea TO'IKYrrepeTHHY AiaroHaJieH Tpa- rre~il. AiJIHTb OAHY ocHoBy y BiAHOIIIeHHi m : n. B .llKOMY BiAHO-
IIIeHHi BOHa AiJIHTb APYrY OCHOBy?
24. Y Tparre~il ABCD a AiarottaJIJIIO AC KYTH ABC i ACD piBHi. 3HaHAiTh AiaroHaJib AC, .11Km;o OCHOBH BC i AD BiAIIOBiAHO AOPiBHIOIOTb 12 M i 27 M.
25.JliHi.11, rrapaJieJibHa OCHOBaM Tparreaji, AiJIHTb OAHY 6i'IHYCTOpoHy y BiAHOIIIeHHi m : n. B .llKOMY BiAHOIIIeHHi AiJIHTb BOHa APYrY 6i'IHYCTOpoHy?
26.IIPOAOBateHH.11 6i'IHHXcTopiH AB i CD Tparreaji ABCD rrepeTH-
HaIOTbC.11 B TO'l~i E. 3HaiiAiTh cTopoHH TPHKYTHHKa AED, .11Km;o AB = 5 CM, BC = 10 CM, CD = 6 CM, AD = 15 CM.
27.3HaiiAiTh BHCOTY TpHKYTHHKa AED i3 aaAa'li26, orrym;eHy Ha cTopoHy AD, .11Km;o BC= 7 CM, AD= 21 CM i BHCOTa Tparre~il
AOPiBHI06
28!',D;iaroHaJii Tparre~il rrepeTHHaIOTbC.11 B TO'lajE, a rrpOAOBateH- H.11 6i'IHHXcTopiH - y TO'l~i F. ,lJ;oBeAiTh, m;o rrp.11Ma EF AiJIHTb OCHOBH Tparreaji IIOIIOJiaM (MaJI. 255).
29tr Y piBHo6eApeHOMY TPHKYTHHKY ABC a ocHOBOIO AC i rrpoTHJiemHHM KYTOM 36° rrpoBeAeHo 6iceKTpilcy AD.
1),lJ;oBeAiTh rroAi6HiCTb TpHKYTHHKiB ABC i CAD.
2)3HaiiAiTh ocHOBY TpHKYTHHKa ABC, .11Km;o H:oro 6i'IHacTopoHa AOPiBHI06 a.
30.KyTH B i BI TpHKYTHHKiB ABC i A 1B IcI piBHi. CTOpOHH TpHKYTHHKa ABC, m;o rrpHJI.RraIOTb AO KyTa B, y 2,5 paaa
6iJihIIIi |
BiA |
CTopiH TPHKYTHHKa |
A 1B1Ci, |
m;o rrpHJI.11ra10Tb |
AO KyTa |
B1. |
3HaiiAiTh AC i A1C1, |
.11Km;o |
ix cyMa AOPiBHI06 |
4,2 M. |
|
|
|
|
31. Y TpHKYTHHKY ABC a rocTpHM KYTOM C rrpoBeAeHo BHCOTH AE i BD. ,lJ;oBeAiTb, m;o !::,. ABC= !::,. EDC.
32tr Y rocTpOKYTHOMY TPHKYTHHKY ABC rrpoBeAeHo BHCOTH AD, BE, CF. 3ttaif:AiTh KYTH TpHKYTHHKa DEF, attaIO'IHKYTH TPHKYTHHKa ABC (MaJI. 256).
33:'<,lJ;oBeAiTb, m;o 6iceKTPHCH TPHKYTHHKa DEF y aaAa'li32 JiemaTb Ha BHCOTax TpHKYTHHKa ABC.