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Киевский национальный лингвистический университет
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9 KJiac

190

0

0

e

MaJI. 289

Man. 290

PaiJianno10 Mipo10 KyTa Ha3HBa6TbC.H Bi,D;HomeHH.H ,n;oBmHHH Bi,D;- noBi,n;Hoi ,D;yrH AO pa,n;iyca KOJia. 3 <l>opMyJIOIO ,!l;JI.H ,ll;OBa<.HHH ,n;yrH KOJia BHIIJIHBa6, w;o

l

n

n,

Ii=

180.

ro6ro paiJiaHny Mipy Kyra iJicra10rb 3 zpaiJycno/, MHO:JICeHHRM Ha

1; 0 0 • 3oKpeMa, pa,n;iaHHa Mipa

KyTa 180° ,n;opiBHI06 n, pa,n;iaHHa

.

n

Mlpa Ilp.HMoro KyTa CTaHOBHTb

2 .

O,n;HHHI~e10 pa,n;iaHHOi MipH KyTiB s paiJian. KyT o,n;HH pa,n;iaH - n;e KyT, ,ll;OB)l(HHa ,n;yrH .HKOrO ,ll;OpiBHI06 pa,n;iycy (MaJI. 290). rpa,n;yc-

Ha Mipa KyTa

.

B O,ll;HH pa,n;iaH ,n;opiBHI06

1800 ~ 57°.

 

 

 

 

 

 

n

 

 

 

3 a A a q

a

(50). 3Ha:H,n;iTb pa,n;iaHHY Mipy KyTiB TPHKYT-

HHKa, sKru;o LA= 60°, LB= 45°.

 

A ,n;opiBHI06

 

Po a B'sa a H H s. Pa,n;iaHHa

Mipa

KyTa

60° · 1; 0 0 =

;

. Pa,n;iaHHa Mipa

KyTa

B ,D;opiBHI06 45° X

 

n

 

n

 

 

 

 

 

X

1800 =

 

4

.

3a TeopeMoIO npo

cyMy

KyTiB

TPHKYTHHKa

 

 

 

 

 

L C= n-~-~= ~

 

 

 

 

 

 

3

4

12 •

 

?

 

 

 

KOHTPO~hHI3AilHTAHHH

 

•

 

 

 

 

 

 

 

 

1.IIJ;o TaKe JiaMalia, ,n;oBmHHa JiaMaHoi?

2.,ll;oBe,n;iTb, w;o ,ll;OBJKHHa JiaMaHoi He MeHma Bi,n; ,ll;OB)l(HHH Bi,n;pi3Ka, w;o CIIOJIY'!a6ii KiHn;i.

3.IIJ;o TaKe MHOrOKYTHHK, OIIYKJIHH MHOrOKYTHHK?

4.IIJ;o TaKe IIJIOCKHH MHOrOKYTHHK?

5.IIJ;o TaKe KYT OIIYKJIOro MHOrOKYTHHKa IIPH ,n;aHiH BepllIHHi?

6.BHBe,n;iTb <l>opMyJiy ,ll;JI.R cyMH KYTiB onyKJioro MHoroKyTHHKa.

§ 13. MHOI'OKYTHHKH

191

7. IIJ;o TaKe 30BHiurniii KYT onyKJIOro MHOrOKYTHHKa?

8. ,ll;oBe,D;iTL, ~o npaBHJILHHH MHOrOKYTHHK e BnHCaHHM y KOJIO i onHCaHHM HaBKOJIO KOJia.

9.:w;o Ha3HB86TLCSI ~eHTPOM MHOroKyTHHKa? I.J;eHTpaJILHHM KyTQM MHOrOKYTHHKa?

10.BHBe,D;iTL <l>opMyJIH ,D;JISI pa,11;iyciB snHcaHoro i onHcaHoro KiJI npaBHJILHOro n-KyTHHKa.

11.3Haii,D;iTL pa,11;iycH BnHC&Horo i onHCaHoro KiJI ,D;JISI npaBHJILHOro

TPHKYTHHKa, qoTHPHKYTHHKa (KBa,11;paTa), mecTHKYTHHKa.

12• .HK no6y,11;ysaTH npaBHJILHHii onyKJIHH mecTHKYTHHK, TPHKYTHHK, qoTHPHKYTHHK, BOCLMHKYTHHK?

13.,ll;oBe,D;iTb, ~o npaBHJlbHi onyKJii n-KYTHHKH no,11;i6Hi. 3oKpeMa, SIK~O B HHX CTOpOHH O,D;HaKOBi, TO BOHH piBHi.

14.,ll;oBe,D;iTb, ~o Bi,D;HomeHHSI ,D;OBJKHHH KOJia ,D;O iioro ,11;iaMeTpa He 3aJiemaTL Bi,u; KOJia, T06TO O,D;He ii Te CaMe ,D;JISI Bebe KiJI.

15.3a SIKOIO <l>opMyJIOIO o6qHCJISl6TbC$1 ,ll;OBJKHHa KOJia?

16.3a SIKOIO <l>opMyJIOIO o6qHCJISl6TbCSI ,D;OBJKHHa ,u;yrH KOJia?

17.IIJ;o TaKe pa,11;iaHHa Mipa KyTa?

18.qOMY ,u;opiBHIOIOTL pa,u;iaHHi MipH KyTiB 180° i 90°?

 

3A,ll;A'II

1.

,ll;aHo ,11;sa KoJia 3 pa,u;iycaMH R1 i R2 i si,u;cTaHHIO Mim ~eHTpaMH

•d > R1 + R2. qoMy ,11;opiBHIOIOTb Haii6iJILma i HaiiMeHma BiA-

 

cTaHi Mim TOqKaMH x i y ~HX KiJI?

2.

Poos'SimiTb3a,u;aqy 1 3a yMOBH, ~o d < R1 - R2 (MaJI. 291).

3.,ll;oBe,D;iTb, ~o KOJIH BepmHHH JiaMaHoi He JiemaTb Ha O,ll;Hiii npSIMiii, TO ,ll;OBJKHHa JiaMaHOl 6iJibma Bi,ll; ,ll;OB)KHHH Bi,D;pi3Ka, ~o cnoJiyqae li KiH~i.

4.,ll;ose,u;iTL, ~o B 38.MKHeHOl JiaMaHoi Bi,u;CTaHb Mim 6y,D;b-SIKHMH

,D;BOMa BepmHHaMH He 6iJibma Bi,u; nOJIOBHHH ,ll;OBmHHH JiaMaHol.

5.,ll;oBe,D;iTL, ~o B 3aMKHeHOl JiaMaHoi ,ll;OBJKHHa KOmHoi'JiaHKH He 6iJibma Bi,ll; cyMH ,D;OB)KHH pemTH JiaHOK.

6.qH MOme 3aMKHeHa JiaMaHa MaTH JiaHKH ,D;OBJKHHOIO 1 M, 2 M, 3 M, 4 M, 11 M? fiOSICHiTb Bi,D;nOBi,ll;b.

7.,ll;oBe,D;iTL, ~o KOJIH KiH~i JiaMaHOl JiemaTb no pi3Hi 6oKH Bi,ll; ,u;a- HOl npSIMOi, TO BOHa nepeTHHae ~IO npSIMY (MaJI. 292).

8.CKiJILKH ,u;iaroHaJieii y n-KyTHHKa?

9.qOMY ,u;opiBHI06 cyMa 30BHimHiX KyTiB 'onyKJioro n-KYTHHKa,

B3SITHX no O,D;HOMY npH KOJKHiii BepmHHi? .

10.KyTH onyKJioro qoTHpHKY.THHKa nponop~iiiHi ,u;o qHceJI 1, 2, 3,

4.3Haii,D;iTL i'x.

11.,ll;oee,u;iTb, ~o y qoTHPHKYTHHKa, onHCaHoro HaBKOJIO KOJia, CYMH ,D;OBJKHH npOTHJieJKHHX CTopiH piBHi.

12.CKiJibKH CTOpiH Mae npaBHJlbHHH MHOrOKYTHHK, KOJKHHH 3 BHyTpimHiX KyTiB SIKOro ,u;opiBHI06: 1) 135°; 2) 150°?

9 Knac

192

A,

Man. 291

Man. 292

13.CKi.n:bKH CTopiH Mae npaBHJihHHH MHOrOKYTHHK, HK~O KO>KHHH ia 30BHimHiX :Horo KyTiB p;opiBHIOe: 1) 36°; 2) 24°?

14.,z:t;oBep;iTh, ~o B3HTi qepea O,ll;HY BepmHHH npaBHJihHOro 2n-KyT- HHKa 6 BepmHHaMH npaBHJihHoro n-KYTHHKa.

15.,z:t;oBep;iTh, ~o cepe,D;HHH CTopiH npaBHJihHOro n-KyTHHKa 6 BepIDHHaMH iHmoro npaBHJihHoro n-KyTHHKa.

16.BHpaaiTh CTOPOHY an npaBHJihHOro n-KYTHHKa qepea pap;iyc R OilHCaHoro HaBKOJIO HhOrO KOJia i pap;iyc r BilHCaHoro KOJia. 06'1HCJiiThan. KOJIH n = 3, 4, 6.

17- XopAa, HKa nepneHAHKYJIHpHa AO pa,zdyca i npoxoAHTh qepea :Horo cepep;HHy, AOPiBHI06 CTOPOHi npaBHJihHOro BilHCaHoro TPHKYTHHKa. ,z:t;oBeAiTh.

18. y npaBHJihHOro TPHKYTHHKa paAiYC BilHCaHoro KOJia BABi'li MeHmHii aa paAiyc onHcauoro KoJia. )l;oseAiTh.

19.Cropoua npaBHJihHoro BnHcaHoro B KOJIO TPHKYTHHKa AOPiBHI06 a. 3Haiip;iTh CTopouy KBap;paTa, BilHCaHoro B n;e KOJIO.

20.y KOJIO, paAiYC HKOro 4 AM, BilHCaHO npaBHJlhHHH TPHKYTHHK, Ha cTopoui HKoro no6yp;oBauo KBaApaT. 3uaiip;iTh paAiyc KoJia, OilHCaHoro HaBKOJIO KBaApaTa.

21.Ki.Hen;i. BaJIHKa AiaMeTpoM 4 CM o6nHJIHHo y BHrJIHAi KBaApaTa. BHaHa'ITe, HKHH Haii6iJihmHii poaMip Mome MaTH cTopoHa KBaApaTa.

22.KiHen;i. rBHHTa raaoBo1 aacyBKH Mas npaBHJihHY TpHrpaHHY cl>opMy. HKHH Haii6iJihmHii poaMip Mome MaTH KomHa rpaHh, HK~O AiaMeTp n;HJiiHAPH'IHOlqaCTHHH rBHHTa AOPiBHI06 2 CM?

23.)l;oBeAiTh, ~o CTOpOHa npaBHJihHOrO 8-KyTHHKa o6'1HCJII06TbC$1

aa cl>opMyJIOIO a 8 = R .J2 - -J3, Ae R - paAiyc onHcaHoro KOJia.

24. )l;oBeAiTh, ~o CTOpOHa npaBHJihHoro 12-KyTHHKa o6'1HCJII06Th- C$1 aa cl>opMyJioIO a12 = R.j2 - -J3, Ae R - paAiyc onucaHoro

KOJia.

25!' 3Haiip;iTh CTOPOHH npaBHJlhHOrO n'HTHKYTHHKai npaBHJihHoro 10-KyTHHKa, BnHcaHHX y KOJIO pa,zr;iyca R.

§ 13. MHoroKyTHHKH

193

26. CropoHa npaBHJihHoro MHOl'OKYTHHKa,u;opiBHI06 a, a pa,u;iyc onHcaHoro KOJia R. 3Haii,u;iTh pa,u;iyc BnHcaHoro KOJia.

27.CropoHa npaBHJihHoro MHOl'OKYTHHKaa, a pa,u;iyc BnHcaHoro KOJia r. 3Haii:,u;iTh pa,u;iyc OllHCaHOl'OKOJia.

28.BHpaai'i'hCTOpOHY b npaBHJihHOl'OOllHCaHOl'OMHOl'OKYTHHKa qepea pa,u;iyc R KOJia i CTOpOHY a npaBHJihHOl'O BIIHCaHOl'O MHOl'OKYTHHKa3 Ti6IO caMOIO KiJihKiCTIO CTOpiH.

29.BHpaaiTh CTopoHy a npaBHJihHoro BnHcaHoro MHoroKyTHHKa 11epea pa,u;iyc R KOJia i cTopoHy b npaBHJihHoro onHcaHoro MHOl'OKYTHHKa3 Ti6IO CaMOIO KiJihKiCTIO CTOpiH.

30.BnHmiTh y KOJIO npaBHJihHHH 12-KyTHHK.

31.OnHmiTh HaBKOJio KoJia npaBHJihHHH TPHKYTHHK, KBa,u;paT, npaBHJILHHH BOChMHKYTHHK.

32.Pa,u;iycH BnHcaHoro i onHcaHoro KiJI o,u;Horo rrpaBHJihHoro n-KyTHHKa ,u;opiBHIOIOTh r 1 i R i. a pa,u;iyc BnHcaHoro KOJia ,u;pyroro rrpaBHJihHOl'On-KyTHHKa ,u;opiBHI06 r2. 'lloMy,u;opiBHI06

 

pa,u;iyc onHCaHOl'OKOJia ,u;pyroro n-KYTHHKa?

 

33.

IIepHMeTpH ,u;uox rrpaBHJihHHX n-KYTHHKiB ui,u;Hoc.HThCH, HK

 

a : b. SIK Bi,lJ;HOC.HThC.H pa,u;iycH 'ix BIIHCaHHX i

OIIHC8HHX

 

KiJI?

 

34.

06'1HCJiiTh,u;oBmHHY KOJia, .HK~O pa,u;iyc ,u;opiBHI06:

1) 10 M;

2)15 M.

35.Ha CKiJihKH 3MiHHThC.H ,lJ;OB)l(.HHa KOJia, .HK~O pa,u;iyc 3MiHHThC.H Ha 1 MM?

36.3Haii,u;iTL Bi,u;HomeHH.H nepHMeTpa npaBHJihHOl'O8-KyTHHKa ,u;o

,u;iaMeTpa i nopiBH.HHTe iioro a Ha6JIHmeHHM aHa11eHH.HM n.

37.Poau'.HmiThaa,u;a11y 36 p;JIH rrpaBHJihHoro 12-KyTHHKa.

38.3Haii,u;iTh pa,u;iyc aeMHO'iKyJii, BHXO,Zl;.H'IH3 TOl'O,~o 1 M CTaHOBHTh O;J;HY 40-MiJihHOHHy 'laCTKY,u;oBmHHH eKBaTopa.

39.Ha cKiJihKH 6 rro,u;oumau aeMHHii.eKBaTop, .HK6H pa,u;iyc aeMHo'i KyJii a6iJILlliHBC.H Ha 1 CM?

40.n piBHHX KiJI, ~o JiemaTh ycepep;HHi KoJia pa,u;iyca R, ,u;oTHKaIOThC.H Mim co6010 i ,u;o ,u;aHOl'OKOJia. 3Haii,u;iTh pa,u;iycH ~HX KiJI, .HK~O 'ixKiJihKiCTh ,u;opiBHI06: 1) 3; 2) 4; 3) 6 (MaJI. 293).

41.Poau'HmiTh rrorrepe,u;HIO aa,u;a11y, HK~o KOJia JiemaTh rroaa p;aHHM KOJIOM.

42.IIIKiB Mae y ,u;iaMeTpi 1,4 M i po6HTh 80 o6epTiB aa XBHJIHHY. 3HaH,ZJ;iTh lliBH,lJ;KiCTh TO'IKHHa 060,u;i mKiBa.

43.3HaH,ZJ;iTh ,Zl;OB)l(.HHY ,u;yrH KOJia pa,u;iyca 1 CM, .HKa Bi,lJ;IIOBi,u;ae

~eHTpaJihHOMy KyTy: 1) 30°; 2) 45°; 3) 120°; 4) 270°. 44.· CKiJihKH rpa,u;yciB MiCTHTh ~eHTpaJihHHii KyT, .HK~o ui,u;rroui,u;Ha

1

1

1

1

2

3

HOMY ,u;yra CTaHOBHTh: 1) 3; 2)

4; 3)

5; 4)

6; 5)

3;

6) 4

KOJia?

45. SIKHH KYT yTuop1010Th pa,u;iycH 3eMJii, npoue,u;eHi ,u;o ,u;uox TO'IOK Ha 'i'i IIOBepxHi, Bi,l1;CT8HL Mim .HKHMH ,u;opiBHI06 1 KM? Pa,u;iyc 3eMJii ,u;opiBHI06 6370 KM.

9 KJI8C

194

MaJI. 293

46. 3a ,11;aHHM pa,11;iycoM R = 1 M aHaiiAiTb ,11;oumuHy ,11;yru, w;o ui,11;- nou~as ~eHTpaJibHOMY KyTy: 1) 45°; 2) 30°; 3) 120° ;

4)45 ° 45'; 5) 60° 30'; 6) 150° 36'.

47.3a ,ZJ;aHOIO xop,ZJ;OIO a 3HaH,ZJ;iTb ,ZJ;OB)KHHY ii ,ZJ;yrH, SIKlll;O rpa,ZJ;yCHa Mipa ,11;yru ,11;opiBH10s: 1) 60°; 2) 90°; 3) 120°.

48.3a ,ZJ;aHOIO ,Zl;OBJKHHOIO ,11;yru l 3Haii,ZJ;iTb iI xop,11;y, HKill;O ,11;yra

MiCTHTb: 1) 60°; 2) 90° ; 3) 120°.

49.

3Haii,ZJ;iTb

pa,ZJ;iaHHy

Mipy

KyTiB: 1)

30°;

2)

45°; 3)

60°.

50.

3Ha:H,11;iTb pa,11;iaHHy Mipy KyTiB TPHKYTHHKa ABC, HKw;o

51.

LA=60° , LB=45° .

 

 

 

 

 

 

3Haii,ZJ;iTb

rpa,11;ycHy

Mipy

KyTa,

HKw;o

iioro

pa,11;iaHHa

Mipa

 

 

1) ; ; 2)

: ; 3) ; ;

5n

7 n

4n

 

 

,ZJ;OpiBHIOS:

4> 6;

5) ls;

6) 3·

 

§14. IlJIOID;I «l>IrYP

121.IIOHHTT.H IIJIOID;I

reoMeTpH'IHYcl>irypy Ha3HBaTHMeMO npOCTOIO, HKill;O ll MO)KHa poa6HTH Ha cKiH'leHHyKiJILKiCTb nJIOCKHX TPHKYTHHKiB. Hara,11;a- 6MO, w;o nJIOCKHM TPHKYTHHKOM Ha3HBaIOTb CKiH'leHHY 'laCTHHY IlJIOill;HHH, o6MemeHy TPHKYTHHKOM (MaJI. 294).

IlpHKJia,ZJ;OM npocToi cl>irypH 6 onyKJIHH IlJIOCKHH MHOrOKYTHHK. BiH poa6uBaSTbCH Ha nJiocKi TpHKYTHHKH ,11;iaroHaJIHMH, npoBe,ZJ;eHH-

MH 3 6y,ZJ;b-$1KOl iioro BepWHHH (MaJI. 295).

 

y

~bOMY naparpacl>i poarJIH,ZJ;aTHMeMO TiJibKH IlJIOCKi MHoroKyT-

HHKH,

TOMY IlOBTOpIOBaTH KO)KHHH paa •IlJIOCKHH• He

6y,11;eMO.

,ll;aMO 03Ha'leHH$1IlJIOill;i ,Zl;JIH npoCTHX cl>iryp.

 

,ll;JIH npocTHX cl>iryp nJto~a - ije iJoiJarna BeJtuttuna,

ttUCJtOBe

3Hattennsi RKOi Mae raKi 6JtaCTU60CTi:

 

1)PiBni ljJizypu MalOTb piBni nJto~i.

2).HK~o ljJizypa po36ueaerbcsi na ttacrunu..; ~o e npocruMu ljJizypaMu, ro nJto~a ijiei ljJizypu iJopiBn10e cyMz nJto~ ii ttacrun.

§ 14. IlJiolaj clliryp

195

MaJI. 294

MaJI. 295

3) ll.n.ol."1a KBaiJpara ia croponmo. l."10 iJopienroe oiJunuiji 8UM.ipK>BaHHSI.. iJopitm.roe oOUHUiji.

HKm;o KBaApaT, npo SIKHii i.AeTbCSI B oauaqeuui, Mae cTopoHy 1 M, TO nnom;a 6y,1.\e B KBa,1.\paTHHX MeTpax (M2). HKm;O CTOPOHa KBa,1.\paTa 100 M, TO nnom;a 6y,1.\e B reKTapax. HKID;o cropoHa KBaApaTa 1 KM, TO nnom;a 6yAe B KBaApaTHHX KiJIOMeTpax i T. A·

122. IIJIO~A IIPHMOKYTIUIKA

BHaHa'<IHMOnJiom;y npSIMOKYTHHKa ia CTopouaMH a, b. ,!J;nSI ~ro cno'<laTKy,1.\0Be,1.\eMo, Iilo 11'.JIOWABOX npSIMOKYTHHKiB 3 piBHHMH OCHOBaMH BiAHOCSITLCSI TaK, SIK lX BHCOTH.

Hexaii ABCD i AB 1C1D - ABa npBMOKYTHHKH ia cniJILHOIO

OCHOBOIO AD (MaJI. 296, a). Hexaii S i

S 1 - Ix nnom;i. ,IJ;ose,1.\eMo,

m;o

~. =

~•. Poai6'eMoCTopouy AB npSIMOKYTHHKa ua seJIHKY

.

.

.

 

•

AB H

u

KlJILKlCTL

n plBHHX qaCTHH, KO)l(Ha 3 SIKHX AOPlBHIOG

n-·

exaH

m -

qHcno ToqoK noAiJIY, SIKi nemaTL Ha cTopoHi AB1. ToAi

 

 

(~8) m ~ AB1 ~ (~8)(m + 1).

 

 

3si.AcH, nop;iJIHBmH Ha AB, AiCTaHeMo:

 

 

 

 

.!!!:.

~ AB1~ .!!!:.

+ _!_,

 

 

 

 

n

"""AB"""n

n

 

 

IIposeAeMo qepea TO'<IKHno,1.\iJI:y npSIMi, napaneJILHi OCBoBi AD. BoHH poai6'10TLnpSIMOKYTHHK ABCD Ha n piBHHX DJ>$1MOKYTHH-

KiB, KO:>KHHii a SIKHX Mae nnom;y ~. IIpSIMOKYTHHK AB1C 1D MicTHTL n

nepmi m npSIMOKYTHHKiB, noqHuaIOqH 3HH3y, ii MiCTHTLC.H B m + 1 np.a:MOKYTHHKax. ToMy

(=)m ~ S 1 ~ ( =)(m + 1).

(**)

297.
AE.

9 KJiac

196

8 ........---- [

 

a

a

81. ----.. c,

 

 

 

 

 

 

 

 

 

 

 

'

 

 

 

 

A

 

0

 

 

 

 

 

 

~I

 

 

 

a)

 

 

 

 

 

 

1

6)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MaJI. 296

 

 

 

3

·

v

(

*

)

•

(

**

)

6a'IHMO, ~o 0

6ffABa

AB 1

• S1

HeplBHOCTeH

 

 

1

 

 

'IHCJia AB

1 s

MiCTHTbCH Mim 2!:

i 2!:

+ ~. ToMy BOHH BiApi3HHIOTbCH OAHe BiA

 

 

n

 

 

n

 

 

 

n

 

 

 

 

OAHOI'OHe 6iJibme Him Ha ~. OcKiJibKH n MOmHa B3SITH SIK 3aBrOAHO

 

n

 

 

 

 

BeJIHKHM, TO

.

.

S1

AB1

v

~e Mome 6YTH TlJlbKH

TOAl, KOJIH

s =

AB ' ~o

H

Tpe6a 6yJio AOBeCTH.

Bi3LMeMo Tenep KBaApaT, HKHH e OAHHH~e10 nJio~, npHMO-

KYTHHK i3 CTOPOHaMH 1, a

i npHMOKYTHHK i3 CTOpOHaMH a, b

(MaJI. 296, 6). IIopiBHIOIO'IH'ixnJio~i, 3a AOBeAeHHM MaTHMeMo:

S '

a . S

b

T= TI -g;= T·

IIepeMHOHtHBmH ~i piBHOCTi, AiCTaHeMo:

8= ab.

0Tme, n.1'0Uf4 npsiMOICYTHUKa i3 CTOpoHaMU a, b o6'1f.UC.1'10eTbC1' aa tJ>opMynmo S = ab.

i2a. IlJIOill;A IlAPAJIEJIOrPAMA

Hexa:H ABCD - AaHHH napaJieJiorpaM• .HK~o BiH He e npHMOKYTHHKOM, To OAHH 3 :Horo KyTiB A a6o B - rocTpHii. Hexaii, HanpHKJiaA, KYT A - rocTpHii, HK 3o6pameHo Ha MaJIIOHKY

OnyCTHMO 3 BepmHHH A Ha rrpHMY CD nepneHAHKYJIHP IIJio~a Tpane~'i ABCE AOPiBHIOe cyMi IIJIO~ rrapaJieJiorpaMa ABCD i TPHKYTHHKa ADE.

§ 14. llJiow;i <l>iryp

OrrycTHMO TaKom rreprreHAHKym1p

BF 3 BepmHHH B Ha rrpHMY CD. ToAi rrnorn;a Tparre~ii ABCE ,u;opiBHIOS cyMi

rr.riorn; rrpHMOKYTHHKa ABFE i TPHKYTHHKa BCF.

IIpHMOKYTHi TPHKyTHHKH ADE i BCF piBHi i TOMY MaIOTh piBHi rrnorn;i. 3BiACH BHIIJIHBas, rn;o rrnorn;a rrapanenorpaMa ABCD AopiBHIOS rrnorn;i rrps.1MoKYTHHKa ABFE, T06To AOPiBHros

AB ·BF.

197

Ae------

4B

f

0

F

[

 

MaJ1. 297

 

 

BiApiaoK BF Ha3HBaSThCH eucoro10 rrapanenorpaMa, HKa BiArroBiAas CTopoHaM AB i CD.

0Tme, n.n.ozq.a napa.n.e.n.ozpaMa iJopi61uoe iJo6yncy uozo cToponu na 6ucoTy, npo6eiJeny iJo 1fie'i cToponu.

124. IIJIOID;A TPHKYTIDIKA

Hexaii: ABC - AaHHH TpHKYTHHK D (MaJI. 298). ,lJ;orroBHHMo ~eii: TPHKYTHHK

AO rrananenorpaMa ABCD, HK rroKaaaHo Ha MaJIIOHKy. Ilnorn;a rrapanenorpaMa AOPiBHros cyMi rrnorn; TpHKYTHHKiB ABC i CDA. OcKiJihKH ~i TPHKYTHHKH piBHi, TO rrnorn;a rrapanenorpaMa AOpiBHIOS IIOABOSHiH rrnorn;i TpHKYTHHKa ABC. BHcoTa rrapanenorpaMa, HKa BiAIIOBiAas cTopoHi AB, AOPiBHIOS BHCOTi TpH-

KYTHHKa ABC, rrpoBeAeHiH AO cTopoHH AB.

[

A B

MaJI. 298

3BiACH BHIIJIHBas, rn;o

n.n.o~a TpuKyTHUKa iJopi6HJOe no.n.o6UHi iJo6yTKY uozo CTOponu na 6UCoTy, npo6eiJeny iJo 1fie'i cToponu:

S =_!_ah.

2

,lJ;oBeAeMo Terrep, rn;o n.n.o~a TpuKyTnuKa iJopi6n10e no.n.o6uni iJo6yTKY iJ60X 6yiJ&-JiKUX uozo CTOpiH na cunyc KyTa Mi:He HUMU.

Hexaii: ABC - ,u;aHHH

TpHKYTHHK (Man. 299). ,lJ;oBeAeMo, rn;o

S =

+AB · AC · sin A.

IIpoBeAeMo y TPHKYTHHKY ABC BHCOTY BD.

MasMo:

1

S = 2 AC ·BD.

3 rrpHMOKYTHoro TpHKyTmrna ABD: BD = AB · sin a, HKrn;o KYT a - rocTpHii (Man. 299, a), i BD = AB · sin (180° - a), HKIIJ;O

9 KJiac

198

8

8

[

D

a)

15)

 

Man. 299

KYT a - TynH:H (MaJI. 299, 6). OcKiJihKH sin (180° - a)= sin a, TO 1v1sr. 6y,zi;h-SJ.Koro BHna,zi;Ky BD = AB · sin a. 0Tme, nJiow.a TpH-

KYTHHKa S =+AB ·AC ·sinA, w.o ii Tpe6a 6yJio ,zi;oBeCTH.

125. «llOPMYJIA rEPOHA .z:vm IlJIO~I TPHKYTBHKA

3 a A a q

a (29). BHBeAiTb <l>oPMYJIY repoHa 1 A.JIB. nJiow.i

TPHKYTHHKa:

 

 

S = .jp (p - a) (P - b) (P - c),

a+b+c

,zi;e a, b, c -

AOB)KHHH CTOpiH TpHKYTHHKa, a p =

2

niBnepuMeTp.

Po a B'SJ. a a H H s.. MaeMo

S = !ab sin y,

,u;e y - KYT TPHKYTHHKa, npoTHJiemHHH cTopoHi c. 3a Teope-

MOIO KOCHHyCiB

c2 =

a2

+ b2 -

2ab cosy.

 

3Bi,D;CH

 

 

 

 

 

 

a 2 + b2 - c 2

 

 

 

 

 

 

 

 

 

cos 'Y=

2ab

 

A TOMY

 

 

 

2 y = (1 - cosy) (1

+ cosy)=

sin2 y = 1 - cos

 

2ab - a 2 -

b 2 + c 2

2ab + a 2 + b 2 - c2

 

 

2ab

 

•

2ab

 

 

=

c 2 - (a - b)2

(a + b)2 - c2

=

 

 

2ab

•

2ab

1

(c - a

+ b) (c

+ a -

b) (a + b -

c) (a + b + c).

= 4 a2b2

l

 

 

 

-

 

.

 

fepoH A.lleKcaa,zi;piAc&KHA

,ll;8BH&Orpeu,&KHA }"leHHA, ftKHA :iKHB

y I CT. H. e.

MaJI. 300
[

§ 14. IIJiouti «l>iryp

 

 

 

 

 

199

Ba.HBIDH AO yBarH, ~o a + b + c = 2p, a + b -

c = 2p -

- 2c, a + c -

b = 2p - 2b, c - a + b = 2p -

2a,

AiCTaHeMo:

2

,-·-,---~------

 

sin y = ab

-yp(p -

a) (p -

b) (p -

c).

 

TaKHM qHHOM

 

 

 

 

 

 

S =+ab sin y =

,jp(p -

a) (P -

b) (p -

c).

 

126. IIJIOID;A TPAIIE:a;II

Hexaii ABCD - AaHa Tpaneaj..H (MaJI. 300). ,lO.aroHaJib Tpane:a;i'i AC poa6HBae I'iHa ,a;Ba TPHKYTHHKH: ABC i CDA. 0Tme, nJio~a Tpane:a;i'iAOPiBHIOe cyMi nJIO~ :a;ux TpHKYTHHKiB. IIJio~a TPHKYTHHKa ABC AOPiBHIOe +AB· CE, a nJio~a TpHKYTHHKa ACD -

+vc .AF. BHcoTH CE ii AF :a;Hx TPHKYTHHKiB AOPiBHIOIOTb BiA-

cTaHi Mim napaJieJibHHMH np.HMHMH AB i CD. IJ;.H BiACTaHb HaaHBaeTbC.H 6UCOTOIO Tpane:a;i'i.

0Tme, n.11.ozq.a Tpanetfii iJopi61uoe iJooyncy niBcyMu i"i ocno6 na 6UCOTy:

S =a+ b . h.

2

B

F D

0

MaJI. 301

3 a A a q a ( 40). ,ll;oBeAiTb, ~o KOJIH AiaroHaJii qoTHpHKYTHHKa nepeTHHaIOTbC.H, TO nJio~a qoTHpHKYTHHKa AOPiBHIOe IlOJIOBHHi A06yTKY AiaroHaJieH Ha CHHYC KyTa Mim HHMH.

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