MOAvnb 1 . CI4CTEMATU3AU|9 TA ySArAIbHEHH9 ..,' onrng I METoAIB nnlxltvtsrpli $ 1 .1 . Ilpo norirry no6ygoey nnaxiruerpii 1.1.8, f. 1.2,3 rprox roqox Ha npsMift ogna i rinrxa oAna Jrerr{r{rb mix 4eoua inruzuu (axc. II,). flpra qrolry Moxrnsi Tpt4 Br?rnagxrz BBaeMHoro poslrirqeuus AaHsx roqox Ha rlpg- uifi. 1) TouraA rerrtr,rrb milrc B i C, roAi aa arc. III, BC =AB + -l AC, are 1,38 4rvr + 2,72 aw + 1,34 gM. Touy rare poe- uiuqenna HeMorrurrrBe. 2) Toura B .nesczrr uim A i C. AC = AB t BC, ane I,34 ptr + * 2,72 gM, + 1,38 ,qlr. Taxe posrrirqeHHs rorrox rarcorr{ He- MOXInBe. 3) Togi eAnne Mo*cJrr,rBe B3aeMHe poauirqenna rpbox roqor A, B i C raxe: Toqra C .nerrtr.rrr irdnc A i B. fleperiplrmo: AB = = AC + BC, 2,72 sNr = 1,38 ,qM -l 1,34 4ttt; 2,72 +nt = 2,72 arr. BiOnoailu: ro.{xa C JreJrcr{rb nri:x roqxauu A i B. 1.3.A, A. 1.4.8, r. 1.S.Ocri;rsxu ea yuonoro M - nnyrpiurnn rovrca ryra.AOB, ro nporrnirr, OM npoxogurt mirx croponaula xyra AOB. Hexafi ZBOM = ,t,.ro+i IAOM = 2x. 3a aftc. III2 ZAOB = ZAOM + ZBOM,' l5O = 2x +.r, 150 = 3t, * = 50. .BOM = 50", .AOM - 50' . 2 = 100'. Biinoaido: E) 100'. 1.6.3a axc. TII,AB = AM + MB. AM = x cm, BM = (r + 3) cru. 12 = x 1- (r + 3); 72 -- 2x + 3; 2x = 9; x = 4,5. AM = 4,5 ctr; MB = 4,5 + 3 = 7,5 (cu). BiOnoai1o: B) 4,5 cu n 7,5 cu. I)) AM = x cm, BM = (r + 6) cnn. x'f (x ): 6) = 42; 2x -l 6 = 42; 2x = 36; r = 18. AM = la cw, BM = 18 + 6 = 24 (cur). ll) AM = x cttr, BM = 2x cr,tt. x * 2x = 42i 3x = 42; x = 42 : 3; x = 14. AM =14clu, BM =2.L4 = 28 (cu). f) AM = 3x ctttt, BM = 4x c:lr. 3x-t 4x = 42;7x = 42; x ='6. AM = 6. 3 = 18 cltr, BM = 4 - 6 = 24 (cwt). A) AM = x cm, BM = 2x ctl. x t 2x = 42; 3x = 42; x = 42 : 3; t = 14. AM = 14 cw, BM = 2 . 74= 28 (cm). 1.9.3a axciouonIII,: IPOM = IPOA + ZAOM. A\ IAOM = xi ZPOA= x * 4O; rc + xc + 40 = 160; 2t + 40 = 16O; 2x = 120; x = 6O. IAOM = 6O'; IPOA. = 60' + 40' = 100'. E)ZAOM=rciLPOA=x-60: r't x - 6O = 16O; 2x - 60 = 160; 2x -- 22O; r = 110. IAOM = ll0"; IPOA = 110' - 60'= 50'. B) ZAOD = rci ZAOM = O,6tci x * 0,6x= 160; 1,6* = 16O; x= 160 : 1,6; * = 100. /AOP = LOO'; ZAOM = 0,6 . 100" = 60'. f\ IAOM = tci ZPOA = 3xl x I 3r = 160; 4x = 160; r = 40. IAOM = 40'; ZPOA= 3 . 40" = 120". I) ZAOM = 3x; ZPOA= 5x; . 3r 't 5x = 160; 8r = t$Q; a = 20. lAoM = 9..20" = 6o"; ZPOA = 5.20" = 100", 1.1O. a)V e l; P e l; S e P8; W e Pqi B e l; <rouxa S.nexrarr nainr toqrcama P i 8>. 6) R e m; C e mi B e MR; M e -41?; <rouxa X se.nemr,rrr uin< ro.rxanrra/. i M>. l.11.OA=OB + AB; AB =OA- OB = 7,3 - 5,8 = 1,5 (cu). _----.-- Ocxinsnrz ro.rxr,r A i B rexarr xa ognift ninnparriri o no- 6----;---T X rrarr(ciBoro ror{Horo o, ro sonra He poSAiJr,frorr,ca roqKoro O, ro6ro, ro.rxa O He Jrefirr,rrb rvrirx ro.*tarvru A i B. flrcu1o 6 ro.rxa A .nexa.na uint rorrr(aMLr B i O, ro BvxorryBaJracb 6ra pinnicrr OB = OA -l AB. A.te qe lreMorlcJrtlBo, 6o sa ymonoro OA > OB. Orxe, rovra B .nexrarr Minc ro.rxaMr,r,4 i O. BiOnoeila: L,5 cm, 1.12, Iloscnesna gzr. N! 1.2. l) AB = AM + BM; 8 = 3 + 5 - flpaBr.rJrbno. To.rxa M JreJrct4rb rvrinr ro.rxami,rA i B. 2) AB=AM + BM;L9 =7 + 12 - npanu.nrno. To.rxaM relrcrarr rrrix roqrcaMl,IA i B. 3) AM = AB + M B ; 27 = 22 + 5 - [paBrJrBrro. To-rxa M,nexrr,rrr rrrix< ro.rxandla A i B, 4) BM = AB + AM ; 2L = I + 72 - npaBr.rrrbHo. To.rxa A aerxarr uinc ro.tcanrz B i M, 5\ BM = AM + AB; 37 = 2l + 16 - npaBrrJrbno. To.rrca -d. .nencrzrr rvrinr ro.rxarvrra B i M. 6) AM = AB + MBi 33 = 18 +'15 - rpaBr,rrrbHo. Toqxa B .lrelxr,rrr rvrix to.rxairrr,r-4 i M. iiiiiiiil;{il*i IEOMETPI9 Einqxina O. 9. ra iH. Ara,4eMi,{nraE piaenu !{ti 625 -""" - -"-*t- -r--1 \2 \/_ OB T -fi-h Bi4cranefi uinc cycigniuu roqxaMrr Ha o,q[ry Meurue, rlilr{ caMr.rx rorror. 1.7"3 cu.(21 - 1) = 60 ctlc. BiOnoaiOa: E) 60 cu. 1.8. .--y4 A A) AM = x cltt, BM = (x - 2\ crt. x * (x - 2) = 42:' 2x - 2 = 42; 2x = 44t x = 22. AM = 22 cv, BM = 22 - 2 = 20 (cn). 624 fd* reouerptg sinsxina o. 9. ra ix. Axa,qeuirHri piBeHb liilt{ifli;il A 3 4 E 1 6 B 4 3 f 5 2 .4 3 4 A 3 6 B 1 4 B b 2 f 1 4 A 5 2 l:Iexafi ro.rrcr.r -4 , B, M nexanr na oAnift npsrvrift, ro4i rimxr oAHa s Hr,rx Jrerrclrrb ruix gnorta iurtrt4Ml4, i sa a&cioMoro nrzuiprona4na rigpiertin BrrrcoHyerbcs oAHa rr pirnocrefi: flpr.r posn'aeanni 4iei aa4aui Ba Kon{Horo B yMoB A-A cxJraAaerbc.ff piBHsHHf, oc- noBoro ,snoro e pinnicrr - nacriAox s aftcioMr.r III,: AB = AM + MB.
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MOAvnb 1 . CI4CTEMATU3AU|9 TA ySArAIbHEHH9..,' onrng I METoAIB nnlxltvtsrpli
$ 1 .1 . Ilpo norirry no6ygoey nnaxiruerpii
1.1.8, f.
1.2,3 rprox roqox Ha npsMift ogna i rinrxa oAna Jrerr{r{rb mix 4eoua inruzuu (axc. II,).flpra qrolry Moxrnsi Tpt4 Br?rnagxrz BBaeMHoro poslrirqeuus AaHsx roqox Ha rlpg-uifi.1) TouraA rerrtr,rrb milrc B i C, roAi aa arc. III, BC =AB +-l AC, are 1,38 4rvr + 2,72 aw + 1,34 gM. Touy rare poe-uiuqenna HeMorrurrrBe.
2) Toura B .nesczrr uim A i C. AC = AB t BC, ane I,34 ptr +* 2,72 gM, + 1,38 ,qlr. Taxe posrrirqeHHs rorrox rarcorr{ He-MOXInBe.
3) Togi eAnne Mo*cJrr,rBe B3aeMHe poauirqenna rpbox roqorA, B i C raxe: Toqra C .nerrtr.rrr irdnc A i B. fleperiplrmo: AB == AC + BC, 2,72 sNr = 1,38 ,qM -l 1,34 4ttt; 2,72 +nt = 2,72 arr.BiOnoailu: ro.{xa C JreJrcr{rb nri:x roqxauu A i B.
1.1O. a)V e l; P e l; S e P8; W e Pqi B e l; <rouxa S.nexrarr nainr toqrcama P i 8>.6) R e m; C e mi B e MR; M e -41?; <rouxa X se.nemr,rrr uin< ro.rxanrra/. i M>.
l.11.OA=OB + AB; AB =OA- OB = 7,3 - 5,8 = 1,5 (cu). _----.--Ocxinsnrz ro.rxr,r A i B rexarr xa ognift ninnparriri o no- 6----;---T
X
rrarr(ciBoro ror{Horo o, ro sonra He poSAiJr,frorr,ca roqKoroO, ro6ro, ro.rxa O He Jrefirr,rrb rvrirx ro.*tarvru A i B. flrcu1o 6 ro.rxa A .nexa.na uintrorrr(aMLr B i O, ro BvxorryBaJracb 6ra pinnicrr OB = OA -l AB. A.te qe lreMorlcJrtlBo,6o sa ymonoro OA > OB. Orxe, rovra B .nexrarr Minc ro.rxaMr,r,4 i O.BiOnoeila: L,5 cm,
1.12, Iloscnesna gzr. N! 1.2.
l) AB = AM + BM; 8 = 3 + 5 - flpaBr.rJrbno. To.rxa M JreJrct4rb rvrinr ro.rxami,rA i B.2) AB=AM + BM;L9 =7 + 12 - npanu.nrno. To.rxaM relrcrarr rrrix roqrcaMl,IA i B.3) AM = AB + M B ; 27 = 22 + 5 - [paBrJrBrro. To-rxa M,nexrr,rrr rrrix< ro.rxandla A i B,4) BM = AB + AM ; 2L = I + 72 - npaBr.rrrbHo. To.rxa A aerxarr uinc ro.tcanrz B i M,5\ BM = AM + AB; 37 = 2l + 16 - npaBrrJrbno. To.rrca -d. .nencrzrr rvrinr ro.rxarvrra B i M.6) AM = AB + MBi 33 = 18 +'15 - rpaBr,rrrbHo. Toqxa B .lrelxr,rrr rvrix to.rxairrr,r-4 i M.
iiiiiiiil;{il*i IEOMETPI9 Einqxina O. 9. ra iH. Ara,4eMi,{nraE piaenu !{ti 625
-""" -
-"-*t--r--1
\2\/_
OB
T -fi-h
Bi4cranefi uinc cycigniuu roqxaMrr Ha o,q[ry Meurue, rlilr{ caMr.rx rorror.
1.7"3 cu.(21 - 1) = 60 ctlc.
BiOnoaiOa: E) 60 cu.
1.8. .--y4A
A) AM = x cltt, BM = (x - 2\ crt.x * (x - 2) = 42:' 2x - 2 = 42; 2x = 44t x = 22.AM = 22 cv, BM = 22 - 2 = 20 (cn).
624 fd* reouerptg sinsxina o. 9. ra ix. Axa,qeuirHri piBeHb liilt{ifli;il
A 3 4
E 1 6
B 4 3
f 5 2
.4 3 4
A 3 6
B 1 4
B b 2
f 1 4
A 5 2
l:Iexafi ro.rrcr.r -4 , B, M nexanr na oAnift npsrvrift, ro4i rimxr oAHa s Hr,rx Jrerrclrrbruix gnorta iurtrt4Ml4, i sa a&cioMoro nrzuiprona4na rigpiertin BrrrcoHyerbcs oAHarr pirnocrefi:
flpr.r posn'aeanni 4iei aa4aui Ba Kon{Horo B yMoB A-A cxJraAaerbc.ff piBHsHHf, oc-noBoro ,snoro e pinnicrr - nacriAox s aftcioMr.r III,: AB = AM + MB.
1.13. AO = 18 cM' OB = 14 ctn.
ro ,-l do = 9 cu; ona ' lon = 7 *r.Z--- 2
KM = KO + OM = 9 + 7 = 16 (civt)'
Bilnoeidu: 16 cur,
1.14. AO : OB = 3 : 5; AB = 48 cu.3a anciomoro eumiproeanna ni4piaxia AB = AO + OB'
IIosHaquMo xoe$iqient nponopqiituocri l'To4iAO = 3x; OB = 5x; 3r + 5x = 48; 8x = 48; x = 6'
AO -- 3 '6 = 48 (cnt); OB = 5 '6 = 30 (cu)'
Bi}noai1u: 18 cu i 30 cm.
1.15. OCrti.xrx[ TorIRa B uoXe sHaxOAI'ITI4c,fl no o6ugna 6onz ni4 rovxu C' To poBUItl
HeMO ABa Br{na{fiu.
L) Mc=f,ec=rz,z=oMB=MC-BC=6-5=
x--T---i'
llpz reperrHi gBox lpsMux a i b yreoprororrcx napu cyuix-rrux (1 i 2, 2 i 3, 3 i 4, I i 4) ra neprzrcarbHux (7 i 3, 2i 4) xyrin. iix s,'racrzsocri:L Cyma cymixuzx Hyrin gopinuroe 180".2. Beptr.rxarrri xyrr piaui.
t.2'l .72+108'=180'.Bi0noeiOt: E.
llpz neperuni.qBox trp.rMilx a i b ci.rnoro c vrBoprororlc.E napz nyrin:nuyrpiuHi oAnocloponHi - 3 r 6; 4 i 5;uuy:rpiurni pianocroponni - 3 i 5; 4 i 6;nignonigni 2 i 6; 1 i 5;3i7;4i8;rrosHimsi picaocropouHi - 2 i 8; I i 7.,flxrqo npu [eperr4Hi ABox trpfiMfix a i b ciqsoro c eiryrpiuui(sonuinui) pieHocropoHni Hyrrz pieni, a6o cyua nHyrpiuuixo4nocroponnix xyr:in cranonnrr 180", a6o eignonigni Hyrzpisui, ro npaui a i b raparemHi (i naenaxu).
1.22. ZB = Z1; l5 = 17, srt< neprrzxa.nrni.To4i, ocni.nrxz Zl = /7, ro ZB = 15, ane qe eHyrpiurni pianocroponni nyrr nputrpsMlrx a i b i ciqsiii c. 3a ossaxorc naparearHocri upeuux o ll b.Bidnoei}u: a I b.
1 .1 9, Hexaft npaui o, b i c neperunarorl'c;r B r' O' PoarlsneMo
npaui b, c i d'.Bo:r.:rl 3a yMoBoo rar(oxc [eperrluarorbqfl B oA-
nifi roqqi. Ilpunycrzrvro, ilIo q.tr rotrna ni4uinna ni4 rounu O'
ToAi npsui b i c marorr gei cninsHi Toqr{r4' rrlo cynepeqr4Tb
rrac:ri4ry a axcioura Ha.{e}r{Hocri. Otxe, npaui b, c i d raxotx
treperr4Harorbcfl e rouqi O, ro6ro sci 4 npami fleperr4narort'cfl
e oAHifi ro'{qi.
IKOM = tKOC + ICOM =
! ztto".2
g 1.2' Onopxi Sarrr Kypcy nnaHiMerpia
626 {:it rEoMETptg SincHixa 0. 9. ra ix. AxaAervriqHrr piBeHb :{ii;$ii:iil i:ii:iil:ili;,:t FEOMETPIC Einsxina O. 9. ra in. Axa4ervri.rHvri piBeHb iL::l 627
npv npauux -AB i CD i ciqsifi AC. Ortte, AB ll CO.Bi}noaida: AB ll CO.
1.51. Hexafi AM = x cm. ro4i BM =(x - 3) cu aa yMoBorc.BK = BM, AM = AZ, CK = CZ ar ei4pianz Aornqrrux, npoBeAeH[x s o4niei rouxuAo riorra. Tottty AZ = r cM, roAi CZ = (x 4) cu.P"*"=AB t BC +AC=2x-2(x - 31+2(x 4\=2x+2x 6'2x-8=6x 10.
3a yuoeoro 6x -.7O = 28, asiAnr 6r - 38; - = 6:.
AM = Az= o] t"*t. BM = BK = u}- r = af t"*t.KC=cz=u*-n=zj- tcnr).
Croponr.r rpuRyrHHKa:
AB 6r-+B]=92 (cnr. BC=s!*2133333Bidnoeida:9-2 "*, s? .*, 8? "*.333
1.52. [un..ra6.n. ao Nb 1.28.
fi=sJ5 clu; a=ar6=sJ5.16=g @wt); a=2r.1!;d s 9./s 3./3
CK L AD, rogi CK = AB = 6& cw, AK = BC = 8 cM sK rporr4J,ren*ri cropoura npa-moxyrlauna ABCK.
il}:ii:iii:i:l FEOMETPI9 EincHiHa O. 9. ra iH- Aragervri.rxr,rfi pieexu ii:i: 033
V aABE /A = ta0" -nporr,r rcyra 30o).
Bilnoai}a:8Jg cM.
Eaco
E
C
B
D6 cru E 10 cnr
=(r-2)cu.
60'
IKCD = IBCD - 90" = 120" - 90" = 30'.
3 :CKD: KD = cK ' tg 30"; KD = 6J3'* = O t"*1.J3
AD = AI< + KD = 8 + 6 = 14 (mt).
s.^-^ = BC' M .cK =8
+,14 6'6 = G6'6 (cm').22
Bi|noei\t: 6616 cn'.
1,65. Hexafi ABCD - sa"qarra rpaneqis+, BC I AD;BC = 14 cttr, AD = 4O crr.. CE LAD,CE = 39 cM - Br{cora rpaneqii, AC - \i 4iaroua.nr.Ocri:rr,nu EaBKoJro rpaneqii-4'BCD orrucano r{oJro' qe
I]errp roaa, onucalroro HaBKoJro rrpfiMorcyrnoro rpr,rxyrnar{a, Jreficr,rrr, tra cepe-4uni rinoreuyar.
1.84, Cepeguna rinorenyera O 4anoro npsMor{yruoro rpr.rryrura-. Ka ....- qe qeHTp onrrcauoro KoJra, rouy OA = OB = R = 17 cru.AB=17.2=34(clr).3 yABC: AB2 = AC2 * BCz;
Ac =,IAts" - Bc, = $nz - to, = "fu4
- g0XB4 - s0) =
s*" =!nc "" =*.16.80 = 240 (cm2).
Bidnoai1o: 24O ott2.
,14.G4=2.8=16 (crrn).
,{,'ra poar'asanna qiei sagaui cftopucraeMocs MeroAoM nJrorq: nJronly rpr4ryrrr[xaanaft4ervro aa ipopivry.roro lepoxa, a noriu, osaroqu nnouqy i cropony, s (bopMyra
1g^ = =ah anaftAeuo AoBlrcr,rHy sncoru h.'2
1.85. flano: IABC; AB = 29 cu, BC = 25 crvr, AC = 6 cm.BD - nucora, [poBe,qeHa 4o menuroi croporlr AC.Ba Sopuy;roro lepona:
AB+ BC+ AC 29+25+6 60n= , = 2 =t=30 (cu).
BH3AC8
8cu
N(cm).
$11tr1:i.J:li{$:r.:*.:
72 = 5 + 4 + BH; BH = 12* 9 = 3 (cm).
Bt}noailo:1.8
1.76. Bilnoai}a; B) 2, 5 i 6.
a 24 24Jg1.77. R= F= F = ._=8J5 (cm).
Js J3 lJ
Bidnoeila: A) 8G cu.
1 .78. s = s.;Jr. sin 45" = 45J, 'l = 45 lcwr'\-,t2
Bicercrpnca Kyra rp[Kyrur]rrca 4irurr rrporr{Jrextuy cropoHy Ha rigpienn, trpotrop-qifi ni npu,'rer;rraM cropoHaM;
1.81. aABC a rDBE, r"*u #;= #,nB. AC 6 10
= tb (.*).ssinxu AB = on = +
BiAnoBiAb: 15 c:[{.
636 i:1: feOUefRlg SincHiHa O. 9. ra in' AraAeilri'{Heril piBeHb :i:iii:i:i:i::ii: l:*l{iiiliitrji TEOMETPI9 Eincxixa O. 9. ra ix. Axasenirxraft pieenu {{";il 69z
Koop4rznanr a6irarorrca, orrxe, 4iaroHali MP i NK treperuHaTorbcfi i tovrcoro nepe-rrany 4i.namcs naeni.n, A 4e oanavae, ryo MNPK - naparerorpaM.flopinxanuu a6c4ucu roqox N i I{ (eoxz pieni), po6uuo BrrcuoBon, u1o 4iaroua.nr
NI{ raparemua oci op4auar. Aua.norivno, opAunarrz ro,{orc M i P .qopieHrororr 1,2
rony MP OX. Bucuoeon:4.iaroHari napaJerorpaMa trepfleH4hTcyJrspni, a qe oaua-qae, qo MNPK -
30, V aABC ZC =9O", CD - 6icexrpuca /C,AD : BD = 5 : 12. CM - weqiana, CM = 26 cur.Ba snacrr4eicrro 6i€exrpzcz Hyra rpnnyruunanona 4iaurr cropoHy Ha sigpisru, nponopqifini
.ACAD1nprlJlerrtl M cTopoHaM: Atsc BD 12
flosuaqueuru xoe(piqieur nporopqifinocri l, uaemo: AC = 5x, BC = 72x, V nparro-xyrnouy rpnryrnrlxy ABC qenrp orucaroro KoJra - qe cepeAlua rinoreuyeu -4-8,rotty AM = CM = BM = R = 26 cm, AB = 26 . 2 = 52 (cm).Y rABC Ba reopeMoro flitparopa: AB2 - AC2 * BC'; 522 = (5x)2 + {l2x)';25f + L44xz = 27O4; x2 = 16; x = 4 {x > O).
AC = 5 .4 = 20 (cu). BC = 72. 4 - 48 (cu).P^*"= AC + BC + AB = 2O'+ 48 + 52 = 120 (cu).
31. Hexafi -4-BC - panufr, rpraftyrHlrr, AC = 5 cw, BC = 3 cu.CO - horo me4iana, CO = 3,5 em.
,{o6y4yelro a AC B lo raparerorpaMa, npoAoB*cr{Brxt4CO ga roqrty O na ni4pisox OD = CO = 3,5.Poar.nsnemo rCBD, y ,sxoro CD = 2CO = 7 cm,BD = 5 cvt Ba BJracrr{BocrfiMr{ rrapaJreJrorpaMa.3a reopenoro rocrauycieCD2 = CB2 + BD2 - 2CB . BD . cos ICBD,
cB2+BD2-cD2 9+25-49 -15 1cos Z('Bl)
32. V pirHo6ivHifi rpaneqii -ABCD (BC I AD) AB = BC == CD; BD - 4iarona,rr,, CF - nucora, O - roqra ixneperrzHy. CO = 15 cm, OF = L2 cv. a BOC cq a DO.Foa ABoMa.r{yramu (ZCOB = lFOD an neprr,rxa"rrui,ICBO = IFDO st* eHyrpiuui pieuocropogHi npr tra-paJreJrbrnx npsMr.rx BC i AD i ciqnift BD. A
28. Hexaii eucota BD 4i;rurl cropony 4C na ei'4pisru
AD i CD, AB = 2$ cNr, BC = 5 cr$' 1A = 6O" '
B rABD.. BD = AB.sin6o" =&f= 3 (cu)'
3 r BDC: BCz = BDz * DC2 (teopeiua lli(paropa)'
Dc = JEc, - BD, = ^ts,- =,lri-= n4o = a (c}a).
{acrrxa 3
Sna.rens,s r i -R (pa4iycin Bnucauoro fi onucasoro ria auafigemo e tpopuyn 4,rao6.rucreus,fl n.noql rpzryrHurca.
abcS=DrTa S=-'4R
DM
642 lti; reouerptg EincHiHa o. 9. ra in. AxaservrivHur piBenb :i{iiii:i;r* ifiiii;iiii;i TEOMETPIC EincHixa O. 9. ra in. AxaAeuiqnrafr pieexu l!;il 643
3a yuonoro BC = CD = 45 ctn. Ocsint'wuABCD * pinno6ivHa rpaneqia, roAD = 2. FD + KF = 2FD + BC = 2' 36+ 45 = 117 (cu).
s - BC + AD gp = a':lll .27 = 2187 (crsrl.
22
MOAvAb 2. BCryn [o crEPEoMETPli
S 2,1 . Ocxoeni noxqrrn crepeoMerpii. Axciorvlu crepeovrerpii
2.3. B) 1, 3'i 5.
2.9.frrqo n.norqi,rurz c i B neperutarorbcrr tro npfi-nrifi a, n.rrorqtnr 0 i y - no npmrifi b, nprauouynpaui a i b neperr.trarcrscn n rovqi C, ro qnTorrxa EaJrex{r{rr ycim.rpr,ou nnorqr,rnau cr, pi y, a ana.rrzrr,, aa arciouoro II, rourca C laarre-rrnrs i npauifi c neperaHy l,rorqlrn o i y.
E 2.2. Hacnlarr s arcioM crepeouerpii
Teopeua l. {epes nparry i Toqrty, qo He Harer$rrrt, ifi, uoxna trpoBecrrr nroq}r-ny i 4o roro nc riarxia o4ny.Teopeua 2. ,flHrTIo 4ni rovxa npauoi na;rexarr, u.norqraHi, ro i Bcs np,fiMa HaJre-xrrr qifi nnoulnni.Teopeua 3, 9epea rpfi Torrnrr, rqo Ee Ha,'retfiarr uprrvriii, Mo)+cHa npoBecrrr nJrorqu-ny i go roro :rc ri.nrxz o4ny.Ilpra poar'aeaxni crepeouer:pfiqnrzx sa.qaq fleprn oa nce neo6xigno cnpo6yoarznu4i.nuru trJrorqrrHy, rraM caMrrM eniuur,r sa4avy 4o n.nanitrerpuvnoi.Il.norqr,ruy oAnoaHarruo Br{aHat{arorb:1),qni npsui, rqo fieperr{rrarorbcfi;2)nparra i rovra, rrlo He HaJrextratr qifi rparaifi;3)rpra roura, sri ne nexa:rr na ogHift nparuifi;4)4si napa;re;rr,Hi upaui.
2.1O. B, A, E, }rC.
2.11. B) 2 i 3.
2. 1 2. I) Hecrcin.renuy rinrrcicrr.
2.13. I\ MD.
2.14.t) era.2.15. D 1, 3. 6.
2.16. A) (ABC) i.B) (MBC).
2.17. B) fl'arr.2.18.f)1i4.2.19. B, r. '
2.1 ,A) o.qua.
2,4.A) o4na.
2.2. n\ 6egni,a.
2.5. A) o,qria.
2.6,Hexaii a t b * 4ani npami, qo Marorr' cni;trny rouxy A, to,4i aa axciouoro II, ve-
pes Hux MotnHa trpoaecru nJrorurrHy, ao roro r+c ri.nrrcr'r o4uy. Ilpurryctrmo, ulo ic-
Hye Apyra cni.nrua roqxa npflMr,rx a i b - ro'rxa B' To,4i icuye Apyra trJrorqftEa' Iqo
npoxoAr{rb vepee npanni a i b, ane qe cyflepeqzrr, axciorri IIr. Orxce, Harne npfiuy-rrleun.tr HerrpaBumne. Ilpaui a i B neper:r'rnarorbcs lTr'rfire s o,4nifi ro'{qi.
2.7,flpunywulno, rqo rouxr A , B i C ne Jrerftarb ua ogniit npsuift. To4i uepea ,4ni npa-
vi AB i AC Morrcua rrpoBecrrr nJrotr{rlHy, i 4o roro x ri.nsxu o4uy (arcc. IIr).Are qe cynepevurs yltoni, 3a flKoro rovxra A, B i C lerxatr y ftorr(Hifi s 4eox pianrzx
2,8,IJexafr fiJrorqfiHr{ a i p neperunarotics no npsmift a,
. b c ct.3a anciorroro II, nparaa o uicturr' yci cni.nruirovnu n;rorqun cr i B.Hexaft npaua b neperlruae n,'roqtruy g s ro'rqi ATo.ui A ..- cninrna rorrra nJrorqlrrru u i B, A e a'
2,2O, Hexafi nparrri a, b i c npoxo4nrb qepea rovxy O i nepe-rr4uarorr, npauy d oi4noni4no B rorrxax A, B i C.Ilpnrra d i ro.rxa O (O e d aa ymororc) o4uosHatrno Brrarra-qaloTb nrtolqr{Hy cI.lConrsa a flpsMrrx a, b i c v,ae B nroulrrHoro a 4ri cni,ruri rov-Kr, - roqxy O i roury neperrrHy 3 rpsMoro it (A, B a6o C).To4i aa nac.nigxorvr a axciorvr crepeoinerpii (reopeiraa 2) npami a, b, c i d tencarr r og-nift n.norqr,rni, ro6ro ro.rxr A, B, C i O .nexar:r s o4Hiti nloqnui.
2.21. Hexair ABC - 4anrzft rpnryruntt, BD - fioro *re4ia-lla, M - 'rorrxa neperfiHy rre4ian, rovxz A, C i M rrew'arsy n,rorqr,rni 6.Toqna D HaJrerfi r4rr, r'patljfi AC, a sHairrrrb, i n;roquui 6. To4isa reopeMoro 2 upnrra DM netxwrs y nlorqr.rni 6. B e DM,ro,qi B e 6.
2,22. Hexafi a - Ea:eta nprma, 3a asciorvroro I, icnyerouna A, rlo rre JreECr{Tb na npamifi o. Ba reopemoro1 uepea np,EMy a i rovrcy A MorxHa npoBecrra nJrorrlr4 -
ny or. 3a arciomoro I, icirye rouxa B, r{o rre Jrerfir.rrby n:rorqr,rni crr. Ilpore4euo qepes fipsMy a i ro'rry BnJrorqr,rny ctr. Il.norqlrnra o, i o, piaui, ToMy rrlo roqxaB n.norqrauu cx.2 He J'IelrdffTb y n.nouluui cr.
n]a, icnyroro rorrxu, u{o I mi trarors cniar,Hy tovxy, I rqr'IH}t MarcTb crri.lsny rou-
Axciomz crepeouerpii:
HaJre?+(arb uiii naorqilni. I ro .{epe3 IJ14x }lo)rcHa npo- | Ky' To BoHIz treperr'IHa}orb-
i roqmfi, srti He sarexars I necru nroqllHy i 4o roro I ca no npauiii' u{o npoxo-ifi.Ac d,C e o., l x ri,rtHu o4Hy. l Aursvepea ulorouxy.
644 l::I: TEOMETPI9 EincxiHa O. 9. ra ix. AxageuivHnfi piBeHd !tt{li:{:it* itlililii;:t IEOMETPI9 SincxiHa O. 9. ra in. Axaseuirnrafi pieenr tj:i:f 645
r
2.23. la:ai rpffMa a i ro.rxaA Bu3naqarorb nlonqlnry cx (te-operra 1). flrcrqo npaua b npoxogurr, vepea rovxyA i ne-peruuae npsMy a n roqrli B, ro nparvra b Mae s nllotrIunorocr 4ni piani cniarsi rovnu (A i B), rouy Jler+curr y nro-rqzri cr (reopeua 2).
2,24, fl.xr4o ro.rxu A, B, C i D He rlexcarb a oAHift n.norquxi, ro sci sour,I i l*cogui rpuB Hrrx He Jrelr{arr na o4nift npnuift, Tomy xoxua B qorr,Ipbox MorrtJrr{Bl,rx rpifior ro-uor (A, B, C\, (A, B, D), (A, C, D) i (8, C, D) ozaravae eAEHy rJrorr{uuy (teopeua3). Ifi vorzptr nroqtrHr e pisnrnmlr, 6o arrqo 6 6yAr-sni Asi s sux a6irarucs, rorouxra A, B, C i D Jre)fia;ru 6 n o,qnifi n:rorquni, rqo cynepevurr yuoni.Bidnoaidu: 4 n.nonlunr.
2,25, a)V npocropi icnye 6eariu rovox. ui4miuunx ri4 4anoi ro.Jlcl,r.A, Biasmeuo tovrcyB. gepea roqrcrl A i B npoae4ervro npamy. 3a axciorrtoro I, eisrveuo e npocropi rou-xy C, ara He Harrexcrrrb npamifi .43. 9epea nparry AB i rourcy C uoxcsa [poBecrl4eArrHy nJrorqraHy. Ocri.rirru i ro.*cy B, i roqxy C rvrolrtna sa.nrra 6eariqqro cnoco6in,ro i nlorqun vepea rovry A Mo)fiHa flpoBecrn 6eaJriq.6) 9epes Ani Aani roqrcz -4 i B npone4euo eArrHy rp,ffMy AB (axc. Ir). 3a axciolroroI, icnye 6eeri.{ Toqox y npocropi, gxi ne na.nexarr' nparvrifi -A3. 9epea rcor+*ry rarcyrouxy i nparvry-AB Ba reopeMoro 1 rvrorxsa trpoBecr[ nJrolqr4Hy. Orr+ce, vepea gri r:ov-Kil MorfiHa npoBecrn 6eaaiu n.roullrx.n) ,flrrqo 4ani rpra roqxn He Jrer{arrr na o4xifi np'nuifi, To sa reopeMoro 3 uepea uuxMOlrcHa npoBecTrr eAr{Hy rrJrorqrrHy.flxrqo xc rpr{ Toqrr,r Jre}rcarb ua o4nifi nprrvrifi, qepeo ra}tx Mo}Krra fipoBecrrz 6ea,ri.{nJronltrH.
2,26. Hexair ABC D - gauuir uorupurcyrxr{n, giarona-ni axo-ro AC i BD neperanarorrca n rovqi O. 3a axciouoro II,npnui AC i BD, rqo ueperrnarorbcs, oAnoqacHo BI4BHaga-
rorr, nJ'rorqzHy o, Ocrimxr,t npsui AC i BD .ne:rcarr y.n.lro-quni c, ro i roqxr A, B, C i D ;rerxarr y qifi nnorquni.
2.27.IIponegeuo npsrvri AB i AC.3a arciorvroro II, npore-AeMo rrepea Hrrx fiJrorqilHy o. 3a yMoBoro M e AB, K eAC, ro4i rosnu M i K .nen<arr y n,rou4uni cr, a sHaqr4rb,i npana MK aer'wru y n.rorqnui o (TeopeMa 2). X e MK,ornre, X e (!.
2.28. ABCD - pom6, a ll BD.Ilapa;ren xi npnMi a i BD oA-HOsHaqHO BrrSEaqaroTb trJrOrlUEy C[, TO'rKa neperr.rHy gia-rona.neft pou6a O rraJreJrc[Tb uparuii BD, a or:rce, i nro-u4vui o,. Togi giarona,rr /.C poilr6a rraJrexrrrrr, nnorqr.rui o(reopeua 2). Taxlru vr.rxou, yci BeprxrdHri pou6a i upauazr Jrerfiarb e oaHir? naouruni.,{ore4emo MeroAoM ni4 cynporueHoro, rqo npaui CD i a neperunaror6cs. flpianycrr.r-Mo, rr{o BoHr,r He fleperrruarorrca. To4i CD ll o, i vepes rouHy D, rqo ne Jrextllrr' rranpamifi a, npoxoar,rrb 4ei npaui (BD i CD), napa,re.nsni c, rrro cytrepequrr axciouiV (axcionri napa,rerbnr4x ),Bftcxosox: npaui CD i a reperuHaromca.
$2.3. flepepisr
2.3O. B) (B,BD); I) (/rD1); A) (,438,).
2.31 . B) t, 3, 4.
646 lt{: TEOMETPIq Eincxixa O. 9. ra ix. AraAeuri.{Hufi piBeHb :;:iltii::iiiil
2.36. PQ c (SBC), rosi X e BC. 2,37.To,rxu D i M nextart i e n.norqr.rni(AMD), i n nrorquni (CDB), oruce,DM - fi:r.ia [eperr.rny rlux nJrouI{H.
B
iti:::jitii$l: TEOMETP|9 EincHira O. 9- ra in. Axa4eruirxrai piaexu {J:l: 642
l
t^:x: i
I
I
2,34. Toqxa neperuny X nparvror KI i n.ro-ll1'uau (ADC) - qe roqr(a neperr,rnynpamoi ,I{F npauoi DC (DC c ADC);ocxi.nrxr.r X - cni.nrsa rorrxa firorquH(BCD) i (ADC), ro BoHa rrexurl Ha upfi-ltifi reperzry qux n,rorqzu DC.
2.35. EF c (C\CD), nxotlnuu (C.CD)i (ABC) neperuuarorbc.fl no npsuift CD.Or*ce, uryrana rorlrda - r{e roqra ne-perrany 1{ npsMr4x EF i CD.
l
i
i
2.33.8) 2 i 4.
'flxrqo naorqzna rpani Mrroronpa'Hrr*a i n,rorqzna nepepiay rvrarorr ,rsi crinrsi'roqftr{, To BoHrr fieperuHarorbc.a no npamifi, qo npoxoAr4Tb vepea r1i ro\nu (niHianepcmuHg naou4un).flpaua, no srcifi fiJ'roqr.rHa nepepiay neperHHac nJrour,rHy rpaui, Hasrnaerbcs cri-dom ntouqunu nepepi,ty.flo6y4ona repepiaie Muororpangunir ipynryerrcr ga nac;riaxax s axcioM cre-peorrerpii:1. ,fxrqo gri l,roqzulr uarorr 4ni cniltsi Torrnr{, To trpffMa, npoBeAeua vepes qiTOqKr.r, e npfiMoro nepeTr,rHy AaH[x nJrorqrrH.2. fxrqo 4ei napaae.nrni r,rorqrana uepernyri Tperboro, ro npami neperuHy na_pa,re;rrHi.cyrr merogy cnigir noanrae n no6ygoni 4onoui'Icnoi fipfiMoi, axa e ao6parxennarvr.iriHii nepernxy civHol n.norqunr4 s nJ,rotrIr4noro rpani $irypu.v sara.nrnouy Br4rraAlcy ciuna n,ro'4una leperrruae [Jrou-rr4Hy xoxnoi rpaHi nrno-r_o,rpaHrrr{Ka i ronrry 3 rrp,fiMax, rra fircux Jrertarr, pe6pa mrrororpannxxa,II{o6 no6y4ynarn c.ni4r.r Ha rpansx i pe6pax, oorpi6"o,I ) no6y4ysaru roqru [eperr4Hy caigy
" ,pu"",r";
2) gni rburn nepepisy, sxi rerxarr r ognifi rpaai, e,e4naru.
ry,B
2.38. fl;rorqranu (ABBr)i (BCC,) ltarorr.+si cniru-si roqxu B i 81, roMYBB, - myraaa.ninis ne-peTrrEy qr4x nJrorqr,rH.
B. C,a-7"'77"'',"| 1/IV
2"39, y nipauigi ABCD roqxa M - roqxa reperlrfiyrregian rpani rASC. flnou{nna nepepiay, u{o npoxo-
Arrrr' qepea pe6po DC i ro'rxy M, neperunae flJrolqu-
uy (ABC) no npsuiit CM. DK - roqrca neperr.rny npauoi /NCM i nnoqann (ABD)- CiT - //f \Ha nroulr.rHa Ivrac rsi cni.mni / Illtt \roqxw D i K o rpannro -ABD, d( ll/lhA .) nroai DK - riniq ilepernHv \ lll-lrut r/civnoi n.nonquu u i rpaui ABD. \[l'/'-aDCK - u:Yxanuft nePePio. Y
2. 44, TIo6y ay emo nepepia xy 6 a ABC DA rB rC rD, n no -rr{uHoro tlc, fiKa rrpoxoAarr, qepeg Bepurr.rHy B, i tov-xnMiN,geMeAAr, Ne CCr.Ocri,ruxr ro.rfir4 ,8r i N - cnilsHi .4JTff n,tou{Hrr L{
i (BB.C\), ro nparra BrN e np,fiMoro neperrzny 4uxtrJrou{}ru. 3'e4naeilro rorrHra B, i N.Anarori.{ao BrM - npfiMa rreperrruy nJrotquu txi (AAP\), Bnaft4emo caiA npsnoi BrM na nro-nluni nlrmnroi ocnonr ry6a. Ide cni.nsxa rovHarrrrorqrrH (AAP) i (ABC), roMy Mae rre?r(arrr Hanpruiii .4-B - npnmifi treperr{Hy qux nJrorrlr4n.AB^MBt=X.Ana;rori.ruo anafi4euo c.ni4 Y npsrroi BrN na nlorquri nzlxnroi ocnonu xy6a.To.rrlr X i Y -: cni.nsni rorrxrr nJlorquu u i (ABC), roMy flpsMa XY e npauoro ne-peTh Hy rlrrx n,lou{ff H.
XY nAD = K; XY aCD-- L. M i K - cni,rrni toqrcn DrrJrorr{EH a i (AArDr),N iL - trrrorq.un ai(DDrCr).3'e4naemo sigpiexaMra M i K, K i L, L i M.BILKM ruynauufi nepepia.Morxe rpanzrucb, ntro ror{Hrr M i N posrvlirqeni rarcrur{unoM, rrlo rpnrvra XY rre rreperuHae pe6pa Hnxnr,oiocnorz ry6a. V qronry B[naAKy moxua snaitrz rovxyfieperr,rHy Z npawoiXY a npflMorc CD a6o nolygyearuNK 'l BtM IMK ll B,N).
2.45. no6y.ayenro nepepis niparui-pn ABCD rrJrotqrrHoro, qo rrpo-xoAZrE,qepeo rovnu I{, S i P, .qeK e AD, S e AC, P e (ABD). IJo-BHaqnMo ii d. tx . (ACD) = 175.ToqHzl{i P.irexcarr y (/BD). IIpo-ne4emo npauy /{P , KP a AB = L.' Toqru S i .L ,remarr y ({BC).TTpoeeremo SL. a Kl,S - uy-xaNnit nepepie.
V 4pyrorrny BlrnaAr{y ro'rxa P ua n.norqurri(/3D) posuirqerra raftnM qfiHoM, r4o flpfiMaKP i nigpieox 1B He neperunarorr,cx, To4isuaxoAr4Mo cttil, X - Toqrty neperr,rHy nps-mux KP iAB..KP
^AB = X, KP ^
BD = R.flponogr.rmo npauy SX: SX
^ BC = @. r{orr,r-
pr,rxyrnlrx KRQS * uryr<auzft nepepie.
TecrAnc caMoKoHTponlor{acrrana 1
2.8. 3. A. 4. B. 5. A. ', 6. E.
DA
2 .4O , V rty6i ABC DArB ,C ,D , r - O -- rourca neperruy Ara-
- (ArCrD) n (A,B,C,) = A,C,;,' (A,C,D) a (AA,D\ = A,D;
(App)^{ccp)=DCf'aArCrD - utYnauuk nepepis'
2.42.Hepea rouny B i nparay /D Morrcna npoBecrr [irorq[-uy (noana.rrzuo ii a) i ,qo roro x ri.nr,Hrjr oguy' 3nafigeuo,
sK rl,rr rrJroilIl4Ha rleperrrHae rpani i pe6pa ry6a'Ocrinrrcu roqxu A i B aelrtarr y unorqrzni cx, roABca.fl.noulunlr (AAID \) i (BB p J ndpa:re,rrHi, roMy rr'roq?rna ofieperr,IHac ix no napaleasHr'IX [psMr{x: BCr I ADrIllynautii nepepis - ABC.DT
2.43. qepee,qiarona,u -4D, rpani AA.D TD i cepe4zny M pe6-
pa 8.B, uoxua npoBecrrz fiJrorllrduy i 4o roro :x rilsxrao4ny. floaHaurzvo ii q..
Ocrcinrxu ro.rrra A i M e cui;rrur{Mrr AJr'fi rrirolqugcl i (-4BBr), ro AM * rrpfiMa rreperrHy 4I4x rrrorquu'3'eguaervro ro.rxNA i M silpiaxoNr.fl.norqunu (AAp) i (BB1C') napa,te.nsni, roMy nrrorqrrua
ct neperr.rHa€ ix no napa.ne.nbHrax rp.fiMr4x. qepea rovnyM napale.;ruro npa*rifi,4-D, [poBeAeMo nprIlry MN (N -roqfia [eperrrny o pe6porvr BrCr).OcHi.nrxn ro.rrrra N i D, - cni.nrui toqnpr rrJlorqr4H a i (ArBrCr)' to NDr - uprma
neperrrHae nci croponra rpunyrurxayrouraxK(7(e AB)' L(Le BC)iM (M e AC)'
ocrci.nsru TpfiKyTHrrn JreE(r,rTb y ,4eanift n.norquni cx,, MorI(eMo sacTocyBaTu anciomy
BSaeMHOrO poSMlrrIeHHfi TOqOTC Ha nJroil{I',rHr.
flpaua a neperrllrae ni4pieon AB, roiuy roqrcrz A i B ;rexarr yHax signocro A.AraroriuHo, roqxn C i B nentarr y piaulax nlBrrJro-
u{nHax. Ale, bcnilrxu npsMa a nepernuae ni4pi-aox AC, To roqftrr A i C raxox Jrelxar-s y piaurxniennoulunax aigHocno npannoi a. IIe osnaqae'rrlo roqxa C o4novacHo Mae SHaxoAurr'Icfi B ABoxpiauux ni elroulnuax siAuocso npxlroi a' rqo He-
Molr{JrqBo. Haue npznyrqeHH.a He[paBrlJr]'He.Ilparwa a He Mo)r{e neperuHarr't rperro cropouy rprrnyrrruna.
18. Hexafi rpra npami b, c i d tpoxoq.flrb qepea rouxy ? inonignoyroqfiax P,QiR.9epea rpalry a i Torrxy ? moxHa npoBecrr'I fiJrorquuy(nocuavumo ii a) i ao roro lr ri.nsmn orsy.To.rxra P i ? npamoi b Ha,telxarr nJronIXHi oc, ro4i aa
reopeMolo 2 PT c d. Ana,rori'{Ho TQ c ct, TR c a.P € a, Re oc, ro4i PR c ct. Q e PR' rouy @ e g.Bracnoeoft: rourcrl ?, P, Q i R Jrelnarb r ognifi n.nouluHi.
19. Hi.
20, rlepea npami -A-B i ,4C, ulo neperr'rnalorbcfi u tovqiA, uorxH€r nponecrll nJrorqr4ny (.4-BC) i go roro xt ri.nr,-
rcrr oAHy. Toqrru M i N HaJre)fiarr, ni4noni4no flpfiMrrM
AB i AC, a or)fie, rerrtarb y nlouluui cr.
BracnoeoH: rouxu M, N, B i C Jrextarb n oguifi n;ro-qrzni.
21. [si Bepmrnu -A i B rplrxyrnlatta ABC i ro.{ra ne-.perrrrry lregiar M Jrelrcarb y nrouluni a. AMr, BMr,CM, .._ megianu rpr{rcyrHrrxa, AII r BH z, CH
" - ftoro
Brrcorr.r, I{ - roqrca fieperrrHy Brrcor.M e u, M" € cr, roMy MMrc a, Q e MM* C e a.3ri4cu ezrnzsae, qo BC - d", AC c d".
Toqnu A i fI, .ne;xarr y n.norqzni u. H e AE' sri[-cuHea.
29. Hexafi npnui a i b neperranarorr,ca n rovqi O, Ilpama c He
npoxoAr,rrb qepes roqfiy O i neperzuae npami a i b riAuonia-nosroqxaxMiN.Ilpxui a i b aa arciouoro II, ognoanavno BusHaqarorb n;rotrI[-ny cr. Touxn M i N npauoi c Harexarb n:roulnui d,, a or,fie,i npnma c Ha-flexctrrr, n.rrorqtt ni cr,
3O. Hexafi Aec.rrrb fipffMrrx fipoxoAflTb qepeo Toqriy O i neperuHarorb trpsMy a. r{epea
np,fiMy a i rovxy O uontHa trpoBecrz nroulr.rHy o, i 4o r:oro rr ri:nnu oAny (reoperyra 1).Koxcsa a Aecsr[ trpsMr.rx Mae B rrJ'rorlrrHoro o 4ri cni;rrni roqrcr.r: O i rouxy neperlHyB npflMoro a. ToAi aa reopeMoro 2 xorxsa B qux trpsMzx rex[Tb y uorquui ct.
31. Yripani4rKLMN P e LK, Re 1{N, S e ML.Ilo6yAyeno nepepis nipauigu uoqzHoro (P/?S).l) P e (LKM), S e (LKM), roruy PS c {LKM),PS=(PRs) a(LKM).2) P e (LKN), R e (LKN), romy P-R c (LKN), I'P1'r=(PfiS)^(IKl/).Bnaii4eno 4ni croponn (pirypn repepiay: eigpiexz PS i P.R.
Toqxa S - cninrHa roqfia ABox nr'roilInu (PRS) i (rNM).
aC
MK
tA
nepeTI4HaIoTL nP,flMY 4 BrA-
650 r:;:::: fEOMETPIS EincHixa O. 9. ra ix. Axa,4euivHhfi piBeHb l!i;i::;:+i;ii: ili::ili*iil: TEOMETPI9 EincHiHa O. 9. ra ix. Axaseuivxrafi pieenu f:;1 651
r
3.18, ABCD - naparre,rorpaM, a MBCN - rpaneqis'rrlo Marorr' cninrny cropoHy BC. BC ll MN sx ocnosurpaneqii, BC ll AD .alc nporr4Jrelr(Hi croponz napaJle-
JrorpaMa. "lop.i MN ll AD sa o3narcorc napaae.nrnoctinpflMnx y npocropi.
3.2O. SABC - rpuldyrua nipami4a. Touru I{, Z' M, N -cepeArzHr.r ni4piaxin SA, AC, BC, SB niAnoni4Ho.
sA = sB = sc = b, AB = BC - AC = a.V ar4SC KZ - cepegua rinia (ea oanavennau).
"Ioti KL = anaroriquo i NM =
M
3,19. CD - cni.rrna ocnoBa rpaneqii4 A-BCD i PZCD'srti He rexcarr e ognifi r.norqusi. @.R - cepegus liniarpaneqir ABCD; QR ll CD. Asa.;rori'mo, MN ll CD-
Toai Qli ll IWt'l sa oaHaxoro napare,rr'nocri npflMnxy npocropi.
zm =!tn=2
3.22. [).3.23. B) CD;t)AB; [) BC;E\AD.3.24.8\ 2 r 4.
fl.norquna o .rirrlrr, npocrip Ha 4na ninnpo-cropz.,flnrqo roqxuA i B;rexars y piaHzxninnpocropax, ro niApisox.A.B fieperlrHae nJro-rr{rrHy cx.. Hannaxz, axrqo rourz C i D.nerxarr,B oaHoMy nisnpocropi si4uocHo cr, ro nigpiaoxCD ne neperunae rlro nJrorqlrHy.
3.25- B) BiApiaoH.Ocxi.nrrcz rrps:o.tiAB, AC tAD ne-
'r{arb Ha o4Hifi n.nor4zHi, To ro.{-
xu Br, C, i D, - cnirrHi roqrrztrJrorrlrH (ABC) i o. Ornre, nonurelrcarr, ira cninrnifi npauifi quxrrJrorqrrH i yrnoprororr niapiaox.
3.21. Toqrcz A, B, C i D ne ,rerxarr, 6 oAniji naorquui.To,txu M, N, K i Z - cepeArrltu ni4piaide AD' BD,BC i AC, CD AB.3a ossase*:r.flw MZ cepeAHs niHis a.A-DC:
MZ t DC. lwt,=-l oc.-2Ana,rorivno, NK - cepe4xa .irini,s aBDC:
Nx =Loc. NK I DC.2
ToAi sa ogHaxoro MZ ll NK i MZ = NK.flonroprenrra nipxynanna, e rprzxyrguxin ABD t A-BC MN I AB t KZ ll AB, sr.ilv.u
1MNllKZ. MN=ZK---AB.2
Are aa yuonoro CD = AB' toti MN = NK = KZ = MZ.gorupuryrnux MNKZ - poiu6 (nporra;rerxHi cropoHu napalellui, rci cropoHupinni).MK I NZ nr giarona;ri pou6a.
3.26.3a yirronoro n.nonluua cr i nJrorq[Ha rpzr{yrnuNaABC treper[-Harorbcs tro rpfluifi BrC,. BC ll B p r BC = t5 cm, AB, : B rB = 2 : 3.yABC * aABrCr 3a ABoMa xliramr.r (ZA - cnin:ruuir., ZABC == /AB:C, ax ei4noni4Hi trpu trapaJreJrsHrx npsMrx BC i BtCr
!ds=9.22bbaa=-+-+-+--a+0,2222
i civuifi -4-B).
"Ioni M - BC , B.c. = AB1 'BC
=ABl B'C' AB
Bi1noeiOa: f) 6 cn.
AB, BCABr + BB1
2 .15= s =tt ("*)'
3.27 , 3a oBHaqeHHsM KM, MN i I{N - nignoniguo ce-pelHi ainii rpuxyruuxin A tOB r B tOC t i A.OC t"toai KM =IUU, rwN =!a,c,, KN =;4c,
o*,, = Lo,", , )",c,.1r,\c, =
- f,,+u, + B,c, + ,+,c1 = |r* ",r.
Bilnoeidu: +* =:'M,B,c,
$ 3.2. BsaeruHe po3Miu4eHHc npeMoT i nnotr1nH14 y npocropi
Moxnuei raxi nzna4nz BBaeMHoro poeuirqenna [pflMoii i nraorrlaur4 y
Teopeua 3. ,fxrqo oAHa B ABoxi 4pyra npaua rar{ox nepertrHa€
654 ;::# rEOMETPIq Einsxixa O. 9. ra in. AragevrivHrfi piBeHb iii::;i;ii;iiiiii:i ii:;iif;:lltit TEOMETPI9 Eincxixa O. 9. ra ix. Axa.4erri.rxrafi pieenu iil 655
rr{r4ny c, no nparrift PZ. Baywonow ni4pisox KM neperunae nJronlr'IHy cr n :ro'rqi O'
roti O e PZ.yOPK * yOZM aa ABoMa liyraMr (ZPOK = IMOZ sxrep'rraxa;rrni, lK = lM sx rnyrpitrni pianoiroponni npu
PK OKMZ I PK i ciqniri MK):
-- =
- .MZ OM
Hexafi OK = r cM, ro,qi OM = MK - OK = (6 - r) cm.
9= ", 6(6-x)=9x; 36 - 6x=9x;irlx=36; x=2,4.9 6-*
OK = 2,4 cv, MO = 6 * 2,4 = 3'6 (cu).
Vci cnimni rorrxr4 rrJron{r{H (ABC\ i o Jrerrtarl sa ix cnimnifi npsrvrift -4-B (axcio-ua IIr). Ocxfumxn X, e (ABC) i X, e u, ro X, e AB.,{;rr no6y4onn ruynanoi rouxr,r X, na nigpiary AC (prc. a) a6o ua iioro npo4on:rcenni -nponeHi AC (pzc. 6) siarMervo rouxy X i npoeegemo nigpisox XX, (X, e AB).
3.33. Ilapaaersiri npaiui BC i AD oAuooaaqrro Blrorraqa-rorb rrrrorrluay (ABC), axa neperlruae nnoqr.rny cL no
pernuy HerrapaJreJrbHrlx rrpfiM{x CD i AB. Ocri.nrrunpama CD q'u,, ro X - roqxa neperprny npauoi CDi n.notqr,rnu o.
3.34. IAXD * a BXC (BC 1l AD aa yuonoro).
^ADAX'' BC BXIjexafi AX = r cM, roAi BX = AX + AB = (x + 2) cu.qe'= = -:-; 2(t + 2\ = 3x; 2x . 4 = 3x1. x = 4.3 x+2Bi}noeidt:AX=4cw.
3.35. Il.norqrana (ABC) np,fiMonyrHr,rxa ABC D nepermtaetrJronlrrHy cr no npauifi AD. Togl aci cniarxi rouxz,n rouy vr.rc.ni fi ro.rxa X, .nencarr, na npauifi.AD.3a y*ronoro DM : MC = 2 : 3'. To,qi 6a y3araJrrtHeHororeopeMoro (Da.neca DX t XA= 2 : 3. Orxe, ro.rxa XMae 3rraxoAr4Tr,rcr na ri4pisxy AD rax,.iqo6 nraxony-Ba;racb qfi pinnicrr.
3.36. Ilo6ynona orucaHa s sa,4a.ri 3.35.3a ymonoro SM : MR -- 2 : 3.Togi, ocni.nrxr MN ll RT, ro SN : NT = 2 : 3Ba yaaraJrbrreuoro reopeMoro <Dareca. ,SZ = 10 cu."SN = 2x, NT = 3x, ST = 5 ti 5x ='l.D; x = 2.sN= 2.2=4 (cm); NT= 3 2=6 (cm).Bi1noei1t:N?=6cu.
3-37.HexaftAB - patazir ni4piaox. C e AB,AC : CB = 2:3.Ocxi.nncu napa.ne.rrni npnui,44r, B B r i CC r rreperr.raarorr,ri4piaor AB, ro sci BoHE re:rcarb n o4nifi n.norquni, araneperrrnae [Jrorq[Hy o uo nparriii ArBr.3a reopervroro @a;reca, ocxi.nrxz AC : CB = 2 : 3,roi-4rC, tCrBr=22$.To4iA.B, : ArCr= (2 + 3) : 2 = 5 : 2.
flpaua i rrJrorqr.rua MoJ{cyrr, 6yru posuiqeui r upocropi rardr4M trr4roM:a) rpsva Jlenurb y naoqrani;6) npama fleperrrHae rrJrorqusy;n) npnua napare,rbHa n.norqrzni.
3.38. Ocri.nrrcu nparriAB i BC neperuxarorr, rrJrorqr4rry c, ro BoHn ifi ue napa-nenrni.AI3 ll DC, BC ll AD sft flport4Jre]fiHi croponlr naparer'rorpaMa. ToAi ra reoperrorc 3(rro4y;m 3), nxqo npaua AB ueperarrae nJrorqrray cx,, ro i nparua DC ii nepernHae.
;ii:illtl:.i;;{'! FEOMETPI9 EincxiHa O. 9. ra ix. Axa4euiqnlrft piaens ii:j 65V656 :i;i: fEOMETPIS Elncxixa O. 9. ra ix. AxagervrisHrfi piBeHb i{tiii*ff
0
A
3.39. Hexaii yABC .lew"yrrs y nlorquHi a, a tABD -e nrouluui F. ln - ix cnilsna cropoua' rqo HaJrelfir4Tb
npnuifi neperuuy fi,rloqrrn cx' i B. Ilpaua CD neperrrHae
r,roqnHr cr i F siAnosiAHo B roqrcax C i D. Eygr-nxaintra npnivra, trapaJreJrbHa CD, neperzuae nJrorrluHr4
cx, i B aa reopeMoro 3 (rrno4Ym 3).
$ 3.3. l-lapanelbHicrb npcMoT i nnotqxHl't
3.4O. I) Besriq.
3.41. B).
3.42. A) oAHy.
g.43. BPt - npflMa reperr4Hy nroqt4H (ABC) i a. BC ll cx.' roMy BC "ll
Bpr''foai IABC N aA-B,C, 3a ,qBoMa lcyraMlr.
AB BC3a yuonorc ABr: BB, = 3 : 1, ABr= 3r, BBr- ia, AB = 4x'
3.54. Heiafi P@ - riHia neperzHy trJrorqrEu cr i firoil{r4-an (MNK). Ba yuoaoro MN )1 a, rosi MN I PQ. Touy
r MNK aa QPK, esiAnn y! - !!. flxqo KP= r cm,QP PK
roNK=(r+8)cn.t-t
= **t, 1Br=9r | 72i 4x = 72; x, lB.9x
BiOnoaiOa: L8 ctut.
3.55- Hexafi s -
trror{uHa, npoBeAeHaqepea xiHeqr eigpiaxa OA - rouxy O. B e OA,BBl ll AAr Bi4piaorc OA i npani BBri AA1,'rerfiarr, e oAHifi n.norqnHi, flKa reperrr-Hae uorluHy o ro rpauifi OAr,V IOAA, BB1 I AA' romy a BB'O N rAArO.
Buio*r BB' - oB - 04 'A4 oA 04BB,.OA BB..OA-Ll AA, ,_. , ,.(cv);
OB OB,
osxa.reHss. flpaua naaasaerybctr naparNenaaoa nnotu,urti, JlHrrlo BoHa He Mae ,fioA-
uoi cri.nrnoi Toqrcr4 3 Hero.ossaxa. flxrqo npaua, sr{a He HaJrelfir{rr nnoulnHi, [apa.'IeJlr,Ha nrcifi-ne6yAr nps-uiji ga uifi nnorrllrni, To BoHa lapaJrerr'Ha i cauifi l,noquHi.
Hac:rigorc 3. flxrqo np,s-Ma fiapareJrbHa Hoxnifiia n.rorqzn, ,flHi neperfi-Harorbcff, To BoHa [apa-ae.nr,na i npnmifi ix uePe-T14Hy.
Hasriaon 2. Icnye 6ea,rivtrpsMrrx, RHi napa-ne.nrHio4Hiii i rift cauifi nno-rrrr,rni.
15. IABC - aAQD {BC II QD,6o BC ]I a, a QD - Cx i @D C (ABC).AB
= B9, qp =
AQ'ac - (1 F 2)'5
= lb (cnn).AQQDABl
BiAnosiAb: I) 15 cm.
16. AB=lPo"",=4 cm. MN =I^(AB+zK); ZK--2MN -AB=2.7 -4-r0(cu).4 *'," 2'Bidnoeida: A) 10 cu.
3.57. Hexafi SABC - rerpaeAp. flror4rana cx rpoxoAr'rrbqepe3 BeprnfiHy S i nepenuuae pelpaAC i BC ocnosuri4noai4no y ro.ncax M i N' M - cepeguna AC' N -cepeAtrHa BC. ToAi 3a osrraqerrH,fiM MN - cepeAss
nittis IABC i MN ll .A'8. Ocsi;mnu MN c (MNS)'To Ba oauarcoro napa.nearnocri npfiMoi i naolquuuAB ll(MNs).
3.58. Ilapa;re.nrni npami AA, i BBl oArrosrraarHo Br,r3r{aqalorb [JIolqIIHy. BilpisoH /-BHaJrelnrrrE, qiii nnorqlrni, 6o A e AAr, B e BBr.tAOAr* tBoBraa,4BoMa KyraMr{ (zao'd,r-- lBoBl sH Beprr{r{turbgi, lA= lB ax
I$;li,i:,:*X uuvtpiruni c4nocroponni)' 't"A' #,= H= Hi#fiii:::::i'f L\ u = 40 , a4 = ELg =+n =,2 (cv)Fj,:l:::i:::::i:S: BB, OB oB b
29, Poar.nsueuo rrpecreuH,fi n nlorqnni rpuxyrnuxa-A,BC.KM - tinir. neperrHy rrolqr4urr (ABC) s nJlotqr4uoro,[apaJreJrr,Horo npanrift AC (u).K e AB, M e BC, KM = 4 e\r, AC = I cv, MC = 15 cu.AC c (ABC), AC 11 u, rorry /C I KM (sa o3Haxoo trapa-.nearnocri npslroi i n.noqzuu).r KBM.a \ABC (aa gnoua xyrauz),KM BM tnBM=r.MaeMo: != " ,AC= BC; rro3'aqrBr g r+lb'
30. Hexafi -43C - ganuit rprrr(yrxrn; M, N, K - cepeAunr4 cropin.4B, BC, AC nit-noni4no,'rlepee rouru A, B, C, M, N, K npoBeAeMo napa.ne.nrni npanti AA., BBr,cc\, MMf NNf KK|NN, - cepe4n,s .niHis rpaneqii CCPTB.
lJrorqr{Hy, sxa trepertrHae nJrorqrrHy o no npeuifi ArBr,O e ArB'oOAir; aOBBrea ABoMa KyraMlI (ZAOAT-- IBOB, an
neprr.rca.nrHi, 1A = lB.fiIc BHyrpirrrni pienoctopouni nprAAr ll BB, i civniit.4S).4 _ AO _'4,O =?.BB, OB OB, 3
floaHaquMo BO = x cM, ro,qi OA = (.r - 3) cM'
= 13 (cu).
P 30cm A
662 i::i: ;EOMETPIS Sinsnina O. 9- ra ix. ArageuiqHufi piBeHb :i:i:i:i:ii:i:i:i;:: :ilt{t:iit{ TEOMETPI9 Eincxixa O. 9. ra ix. Axageuivxnn pieenr illi; 663
91. ABCD - poM6, a I BD,A e a. flpaua a i npawa BDoAHoBHar{Ho Br,rrtarrarorr, flJrolqnuy. HagseMo ii o. 4 e tt'O e rx (ocHinrr*nO e BD), ro4iAC c cr,6o 4ri ro'rcuAi 0 npnrvoi AC lexa'ru Y nloruuui o.Ceo,,DE0,roMyCDcu.Oexi.nrrrz npnlri a i CD .nexarr, y nlou4Izui cx. To Borlu
ne e mnmo6ixnalru.flprznycrzmo, ryo a i CD napalenrxi. To4i n nlorquHi c uepes rovrcy D trpoxoAurb
ani pieHi np.aui BD i CD, napalelrui np.alrifi a' aile \e HeMoltcJrr{Bo'
Orxe, npaui a i CD nePervuarorr'cfi.
32. Ba yrvlosoro a Ji BD, rouy a i BD nexarr n o4uift n:ro-
uluni. Hasseuo ii cr.
A e o, rovxa neperuny 4iaroua-neft ponda O,rexr{Tr' y nJro-
uluni c, 6o narerxzrt nPauiit BD.
,{ni ro.rxx 4 i O npnuoi AC sa,rerxarl nrorqzxi o' roMy
AC c a (reopenila 2, rvroxytn' 2).\s
Orrxe, pom6 ABCD i flprrMa 4 Jre]+(arr, n o4nifi n.noqrzni, a nparvra b -o3uaqae, rqo npxmi a i b - rtralro6irxni.
b ll c aa osnaxoro rapaJreJrbHocri npruoi i n.ro-qn{u.
4.15" Hexafi nprui a i b urruo6incsi, BN6epeuo Ha npfi-uifi a (axc, I.r) Aoni,'rrHy ro.rxy A i npoBe,4eMo .repesuer upauy br, napa.ne.nruy b, (rlepea rortcy, slla rreJrexc[TE xa 4airifi npamifi, Morr(Ea trpoBecru e4r4ryrp,fiMy, napa.uelrriy 4aHifi.) Ilpami c i b, aagdrorreAr{Hy nJroquHy B, 3a oairarcoro napaJ'reJrbHocri npa-uoi i nron{znra b li B.Ana.noriqso [poBeAeMo npfiMy dr vepea goeinray rou-xy C: a, 1l c.9epee nparri a, i b npore4ervo rrJrorrlriny o. .(ri nparrni a i br, tqo flepernuarorbcs,urorrlr{Hu B ni4noni4no napalersHi AEoM trpEMuM a, i b, rqo neperr4Harorbcs, rrJro-rrlr{Hr.r o, 3a oaHarcorc napa.ne.nrnocri uJrotrtr4u cr ]] 9.
4,16. Hexafi trJrorqr,rur4 cr i [3 napa,re.mni, i a cc{. IlplrrrycrrMo. rqo fipflMa d treperrz-. Ha€ rrJrorr{r4rry p e 4eaxifi To.{qi M, ro4i rouxa M - cninrsa rovxa n.xorquu o i B,
rqo cyreperrrrrr, ymoni cr ii B.Bucnoeon: rrpflMa a He rreper[Hae u.riorqzny B.
4,17. Itonrsi gni npnrvri, nlo fleperr,rnarorrcx, aa arcio-uoro II. saAarcTb rrJroil{rrny.fonegeno, qo (ABC) ll (A'B'C').Ocni:nrcu O - cepe4r.rua rc6xnoro e ni4piamin AA'BBr CCr ro IAOB = aArOBr Ba fiepmoro oaHaftoro(IAOB = ZAtOBt nn reprunanrni).B pienocri rpanyrurzxia BrrrrJrrrBae: ZOAB = lOArBrro4i ea oanaxorc ilapaJreJrbuocri npaunx aa nnor4uui AB ll ArBr,Arra.noriqno ,4oBoArrMo, ao AC li ArC'3a osHaxoro napa.ne.nrnocri fl,'ronlux (ABC) il (A.B.Cr).
4.18, Hexafi O - 4aua rouxa.
Il.norqranz y npoctoPi Mo]ltyrb:a) neper:zHatraca no npa*tiii, srcfiIo BoHtr Marorb o,4ay cninrny rovrey;
6) a6irarrca, .fircilIo Y Hrrx e:
- 4ui cnirsni npalri, u1o fleperrlHalorbcs:
- cninrna nprua i Torrxa., rr]o iii ne Ha:!exurr,;
- rpu cninr,ni TorIKu, rqo He rblfiarr, ua o4nifi npgrvriii;
e) Syrt napa.ne Jrbluvrtt, To6To He wratn rxo4noi cni.nsHoi ro'rxra'
A B B T T3 2 1 2
4.8.8. r, lI.
iilti;litiiii TEOMETPI9 Einnxira O. 9. ra ix. Axaseuivxui pieenr ll:! 665
4. 1 9. gepes Beprnr{rrr4 trapaJreJrorpaMa r -BCD nponelr'rnpami.4-A, 11 BB, ll Cc1 ll DDr ari neperny.nu n.no-
rrlr,rHy c B Toqrcax Ar, Br, Cr, Di.Hexafi O - roqria neperrHy 4iaroualeii, -BCD' O -cepe 1nlr,a B D. Ilpoeegeuo OO I ll B B I ro4i sa reopervroro
@a.neca O, - cepe4una BrDr. Ana"norivHo MotrcHa Ao-
Becrlr, rlo O, - cepe4uua AlCr A\BrClDt - trapaJre-
.2o.HexafrB - 4ana i.r*u t o( - AaHa rrJrorrf'Ila.Ilpone4euo qepes roqxy B n.norquuy B, napane.rrnyulorqrani c.Hexai b - ,4oni.nrua npsMa, rqo npoxoAurb rrepe3 Toq-
xy B napa.nearso ct. Bisrueuo n n.norquni cl 4onillnytouryA i npoeeAeMo qepe3 tounyA i npsuy b rrJrolqu-ny y. To4i nJrorrllna Y fieperunae rrrorrl[Hlt a i B no
rrapaJreJrbH[x np,fiMr4x aribr, ale np.alra b, trpoxoAr4Tb
uepes rouny B, a npsMa b raxorx .nerxuru y nnorquui y
i npoxo4ntr vepee ro'rxy B i sa xac.:rigrom 3 TeopeMr'I
4 napa-ne.nrHa ripalrift or-Togi aa arcciomoro napa.nelr,noeri npaui b i b1 Marorb cniBua,4arrz, rouy b c p, qofi rpe6a 6y.ro Aogecru.
4,2i,fIloqnnr.r o i $ ne MoJ+{yrb neperuHarr,rcff. .flrc6r ninorqznu u i B rra;rz cni.nr,iry
Toqxy, To qepe3 qxr roqrcy trpoxoAn rn 6 Asi nJou-u{Hfl (o i 0), napale,trti nlouluuiy. A qe cynelpeuzrr reopeni 2 (uogy.nr 4).
4.22. l\ Ocxi;rrrrl M i N ea yMoBoro e cepeAuHaMrr pe-
6ep AA, i BBr, ro MN I AB. flpz no6ygoni nepepisvny6a n.norrttoro (MON) BpaxyeMo, rqo O - rovxa A1
4.37. Ocni.nrrcu IMAB + IAQP = 180", a qi nyrz e BHyr-piurHi o4nocroponni nprE rpflMux AB i PQ i ci.ruifi,4@,roAB I PQ.IMKQ = ZMCA, a qi xyrz - ri4uoniAni rpu flpfiMr4xAC i QK i ciqsifi KC, rowty AC I QK.(ABC) il (QPK) sa osHafioro napa.ne.nsnocri rrrorqrrn. A
i:iiiililiitlii TEOMETPI9 EincxiHa O. 9. ra ix. Axa4ervrivxrafr pieeau ;|:!:,i 667
4,42, flpami OA2 i OBz, rr{o neperlrHarorbcfi, BrraHarra-rorb n,'rotqlrHy, sxa neperr,Inae rrJrorr{r{Hrl o i B no na-paJreJrbHr,rx flps Milx A\Br i A2B 2.Ocxfursrcu A)Bt ll A282, ro tArOBrN rA"OB"
a) 3'e4naemo rouny K g ro.rnauu .F' i M, a ix - is roq$orc N, .qe.F, M i N - sia-noei4no cepe,qrrrru BiApisniB AP, CP i DP. To,qi FK ll AB, MK I BC sx cepe4ui niniirpurcyrnuxin APB i BPC.3a oananoro napa,'renrHocri nJrorrltrH (FKM) ) (ABC).6) Hexait M - cepesuna AB, N * cepeAuHa BC. 'foti MK'11 AP, NK il PC sH ce-pe4ui ;rinii aAPB i, tBPC.Korxna a,nap npauux AP i PC ra KM i I{N, rqo neperlruarorbcfi, BrraHarrarcrb Aesr{yn,aorqr,rny. KM ll PA, KN I PC, rony (MI{N) ) (APC\.
9= t :B(x+4\=sx;5 x+4'3.tc+L2=5x;2x=12;x=6.OPr = 6 .*.
4,44.Poss'aaanrss asa,rori.rne 4o Nr 4.17.
4,45. BrarouaeMo AoBeAeHH.rr, uo6y4ynannrr.r rani nlorqnnn.PoarrsneMo rpz nonapuo ur.ruo6ixui pe6pa ry6aABCDAIBLC.D1 AlBr DtD i BQ.flo6ygyel,ro repepiau xy6a n,rorqlrnavu (BrDrC) i (A,BD).I-[i n,nouluux Maror.t 3 ai4noai4nrur.r rpanflMu rcy6a no 4uicni.nrni Toqlifi, a orrfie, fleperr,rHarors qi rpaui no flp.rrMr4x,rqo flpoxoAfiTr, vepee qi To.rxfi. TaxuM trrauoM, ABTD'Ci a BDA, - ruyxani nepepiez,BD ) BrDr CDr I BAr ax 4iarona.ni nporra.nerxnrax rpanefirapa;re.nrnocri nrrorrlrH (Bp1C) ll (BDA).L{i napa.ne;nui fiJrorqlrHlr rigrnnarorr sa mr.rlro6irnnrx npsMux pinni si4pisxu ArB'BC i DDf
4.46. [ni npsui a i b ne e napaJre;r;Ht4Mr.r, 6o r:oAi6 ArA" = BrBr, a ca yMoBolo noirrz piaHi..flrrqo npnrvri a i b neperunarorrc.fl, To rrepec Hr{x Mo}r{EaflpoBecrrl urorqr4uy, sra fieperr{Hae ir.rorqunra cx, B i y norrapaJreJrbHr,rx flpffMrrx ArBr A2B2i ABB,.
4A, = Arn,
.ArA" BrB"
62B,B" 5'
(
ry6a. To4i sa ooHar{ox)
3a reopelroro @a,reca:
- B.B^ 23a vMoBoro:' BrB'^ 5
BrB" = (clt); B,B" = 6 + I"5 = 21 (cu).5.6
- = lt)
2
h
/A. L)",i/t
/ o,l-?:,/1f)u1,""1
668 iii: feOugfplg Sincxixa O. 9. ra ix. AragerurivHhfi piBeHb :i:::li:i:::::li::: :ilil{liltil TEOMETPI9 Einqxixa O. 9. ra ix. Axasewrirnnft pieenr fi!: 669
o i B no napaJreJrbnr4x np,tMrrx KH i CP.yKOH * aPOC sa ABoMa ByraMt4 (IKOH = IPOCsK Beprqr(aJrbHi, ZKHO = IOCP ax nnyrpilrni pie-nocroponni npu KH ll CP i ciunift C,FI).
KO _ RH . op _ KO 'CP _ 9,6 . 4,5 = 3.2 (cv).
OP CP . KH 13,5
i
I
I
i
iil
ri
ii
c
9acruxa 1
9. E.
/ ",D:'/
T,L)U,'
18. Ilprui A4,2i Bp2(A4r^ BrB"=C) reperrrHalorbriapa.ne.nriri rrJrorr{r{Ht4 o i B no upaMrrx ArBl i AzB2,ArBr ll ArB2. To4i a,4,CB, N a ArCBr.
94_94ca2- cBz'
. Ocrci.nrxr CA, t Ay', = 1 : 3, ro CAr: CAr= | : 4.
30. Hexaii AB 4anzil ni4piaorc, Ar i 81 npoexqii roqox A i B Ha n.norquHy c,l) AAr ll BBr. Ilpoee4euo uepes npaui -4-4, i BB, nrorquHy. BoHa reperHe nroqzHy
Hexait O - cepeArrHa BBr, ropi AO o a = C, C e AB.2) tAA,C ," aOB.C (OBt ll AA).
oz-=!nn.=4:2=2 cv.'2AA. A.C
Hexail B,C . x cm, ro.qi Oi,= i,C,8x-2x+18;6;r=18;r=3.BrC = 3 cm; ArC = ArB, * BrC = I *BiOnoeiOa: 12 cm.
illiii{iiiill TEOMETPI9 EinqHixa O. f,. ra in. Axasewrivxnfr pieens fri:ir 675
(ABC) no npauifi.4C, AC ll Apr fl.norqrzna (MNK) ne-perr{Hae rpaus AATDTD no npanrirl KM (4ai cnilrsirouxz). Anaroriqqo (MNK) . (ABC) = MN. Il;toqu-uy (ArBrCr) trJlorquna nepepiay [eperr,IHae no npalrifiKF II MN.Orxe, KM ll AAr, MN ll AC (cepe.que:rini* arADC). BaosHarcoro napa.ne,rrnocri [Jrorr{r{H (MNK) ll (AA.C).
32. Ocri.nrrcu ro.mu A, B, C i D He Jregcarr, na oAnifi nlo-rqvrni, ro npaui -48 i CD - unlro6ixni, a anauurr, .ne-
,rcaTb Ha Aesrcrrx [apaJreJrBHr,rx fiJrorqr,rHax 0 i B.Cuoci6 no6yAoBr4 Tariux rrJrolqrru onncanraft y sa,qaqi4.15.Tper* nnorquna Y napaJreJrbHa xoxnift o npaivrr,rx .43i CD, a sna.rrarr, i n;rou{unam o i 0.r{epee upaui AC i AD ra BC i BD, qo reperl4rrarorbcs,npoBeAeMo nroquHrt.(ACD);y = x4s, MK ll cD; (BCD) n y = NP, NP ll CD,rori MK ll NP.Arra,roriuno, MN I KP. SeiAcr po6uMo Br{cHoBoK: MNPK - uapaJleJlorpaM.
MOAvnb 5. nEPnEHA]4Kv.IlgPHlCTb nPCMl4X I nroult4H y nPOcToPl
$ 5,1 . IlepnengrKynepHicrb npeMt4x y npocropi
,[ri npanri y npoctopi Haar{Baror}'cs nepneru0urcynapH'tt'Mu, ar"ruo Borru fieper?rna-rorrca ni4 [psMraM KyroM.Bnacrunocri neplerrArrrcyJrspnux rrpfiMr{x y rrpocropi1. {epee 4oni.nrny rovxy npsuoi y npocr:opi MorfiHa upoBecrr'r trepnerrAr4rdyJr.Epuygo irei npamy,2. ffrcrqo Ani npgrui, cNi nepetr,rnarorrcs, si{noniAuo napa.nelrni ,4aoM nepneHAtr-nyJrfipnl,rM [p.EMHM, TO BOHI,T rerft flepneuAr{Ky.napni.3. rlepea 6ygr-arcy rorrry flpocropy, rqo He HaJre]lcllrr nprMii, Mo]rcHa fipoBecrrilnp.fiMy, nepneH,qHXyJIspHy Ao Aanoi.4. flrcllo npsnra [ep[eHArr]cyJrspna Ao oAniei B ABox napaJreJrbHilx trpffMrrx i,rexursB Hr{Mr. B o,qnift uouluni, to soua rreprrenArrxyJrfipna i Ao 4pyroi npauoi'
2) B aAoB: AB - ^[Ao\ oB' =,ls' - g = ^lz5 +s6= J61 (civr).
B tAoc: Ac="[Ao\oc, =E.s'=Jrb-% ="60 =s.E (cnn).
B aBodz BC = JoC\ oB" = ^ts" * = ,lrs +86= ,,6T lcnn).
P*" = 16T + rJz + "67
= 5"t, + 2.,6T (cu).
3) B aAoB: Au =,[Ao\ oB' = 17 + t'z = J9+ 16 = 5 (crvr).
B dAoC: AC = {Ao' +oe = J"z - rz = Jg -%= J34 (eru).
B rBoC: BC = .tje *oB, = ,lt, * = J% + 16 = GT rcm).
P*"=5+J3++Jar (cu).
4) 3 r,AoB; Au = ,[Ao\ oB' =.8 + 4a' = al6 (c*).
B aAoc: AC = ^[Ao\ oct = ^ta\ ra' = o.u4o (c*r).
B a BOC: BC = Joc\ oB' = ,16c\ 4a' = a",[g (c*r).
e* = o(JB +.u[o + JG) "..
5,11. Hexafi O iOt- qenrpra nporurerxnlrx rpanefi,43CDi ArBtCrDr xy6a - rorrrlr [eperr,rHy giaroua.neft nna4-parin rpauerl.AO = ArOr AO ll Apr rowy AATOTO - rapaJrerrorpaM(nporrz,rexni cropourr pinni i napa.ne.nrrii).Ocxiarxi,r 6iuai pe6pa rcy6a nepneir4nryngpni Aoftoro ocnon, ro AATOTO - npsMoryrrux, OO, I AOi ool L ArotAna-nori.rno AoBoAr{Mo, rqo OO, L BO i OOr L BOr
5,14. Hexaia ABC\ - gart:rir tpanuntuuii rerpae4p. M -cepeguna AD, K - cePeguna BC.l) vAKD - pieuo6e4peuuir, 6o AK = KD stx ne4ianz(a sHavurr, i nucoru) pionocroponHix rpnnyrnraicin '4-BCi. cDB.MK - we4ia:aa r'A.I{D 3a yMoBoro, a snaqzrb. i eracora:
MK LAD.2) AHa,'roriuHo, rCMB - pieHo6eApesuk,6o MC = MB'MK - v'etiana i eucora: MK L BC.
5,29 . ABCD - npane4ia (BC ll AD , 1A = 9O").I)(ABC),MeLOcxiarrn I -L (ABC), ro MB I AB.Ocxiarxu r:panerlia ABC D npavoxyrHur{a3a yMoBoo, ro AB I BC.Tanunn vlrnom, npaua -AB trepnenAznyJrapHaABoM np.flMrrM MB i BC nrlorrlr4Hu (MBC),a rle osaaqae, rqo,4-B -L (MBC) (e a ymonoro).
5,3O. ABCD - npsmoxyrnux, roMyAD I AB. AK L,(ABC), rovyA]< LAD.flpnma AD nepnenAr{nyrfl pna ABoMtpamuvAK iAB nnoqnuu (,4-BK),nni neperzHalorrca n rovqi A. To4iAD L(ABK) Ba oaHarroxl nepreu,qu-rcy;rapnocri npfiMol i n,roq[Hz.
4ai npaui, ani ueper:zuarorbcfl, flapareJrbHi 4eou npa-MuM nroqnHu BBtDtD.BBtDp 1,4C. Biaouo, ruo flKUlo trprMa nepneH.4r{KyJrfipHa o4uiil a aaox napaJrerb-Hfix trJrorTlfiH, ro BoHa neprreH,qr.rxyrflpna i gpyrifi. Orxe, AC I MNPK.
5.15. A. 5.18. B.
5.16. A.
5.17. B.
5.19. B.
5.20. B.
5.21 . B.
5.22.8.
5.23. B.
5.25. 3a yMoBoro rrz n, [pt4 Ltrr'oMy rn I cr. AoseAenro, rqo i n I tx'
Poar:rsneuo gei npaui c i d' tnoulu]f.u cx, nHi neperzualorbctr' Ocxinr'xn n I cr'
ro m L c i m L d'3a oguaxoro. A ocnilsxra n ll m, ro n L c i n .L d' Togi fipflMa z
nepneHAuxyrlrpHa ABoM npfiMuM c i d nnoqu:rru cr, rrci neper:zuarorbcfi' Suavurl'n I cx (aa osnaroro).
5.32. Hexafi ABCD raparerbrpaM, BK f (ABC),FD L (ABC). Tosi. BK ll DF. AB CD nn npor:u-.nerxni croponu fiapanerorpaMa. flni npami AB i BKlJ'rorrlr4Hn (,43K) napa"remni ri4noei4Ho rpauuu DCi DF nnoryuun (DFC). 3a oaHanoro (ABK\ ll (DFC\.
5.33. Posn'saanns ana.noriqse ao Nh 5.32.
AD
iif;tiiitif* rEoMETPlc Einsxina o. 9. ra ix. Araseuivxui pieenu;!:i:i 629
55.2, flepneHguKytepHicrb npeMoTra nroull4Hll y npocropi
flpama, sna treperr4Hae rrJrorqrrHy, rlaar4Baerbcn nepneu1urcynnpnom go qiei n;ro-
rrlrrnr, slcnlo BoHa neptreHArrr{ynffpua Ao 6yAr,-sxoi npauoi, qo 'nbxrzrr' Ha qifinrouluHi i npoxo4lrs qepes TorJKy ix neperuly.Oasana. flHrqo npama trepneHAZnyJrsprra Ao ABox trp.trMrlx' sri 'nexcatr y n'norqlrriTa rreperrrxarorbcs, ro BoHa nepneuAttny,'rJlpua go 4aHoi trrolrluHu'HaoriAxu. 1. .flrqo nJrolquHa neplen.qflIdyJlfipHa 4o o4niei 3 ABox rrapaleJrbuzx
npffMr{x, To BoHa nepneH.Ilu}tyrfipHa i 4o apyroi npnvoi.2. ,[si npsui, nepileHAnxyrspni ,4o o4uiei fiJrotr{r'rnu' napareJrbHl'
Teoperua no rpr{ repreuAurcylapu (TTII)flxr4o npama, flpoBeneHa na nlou4rai qepes ocuoBynoxz.noi, nepneHAl4xyJ'r.fipna 4o 'ii npoenu,ii' ro BoHa
lepnerrAlrxyJrapna i Ao noxu.noi. I nannaxu: axrqonpffMa, rrpoBeAena qepes ocHoBy [oxI'{JIoi n-a nnqu4zni'nepnerlAr4rcyJr,fipHa Ao noxr4lrol, To BoHa rrepFenAr4rty-
.nnpna i 4o upoexrlii noxz.noi.
A 3
F 1 o
B 2 4
A .dcM B
680 i:::::: FEOMETPIS Einsxina O. L ra in. Axagervri'lHrft piBeHb ii:ii:'.:*itii! $iC.t:;ii:il:.ii rEOMETPI9 Eincxixa O. 9. ra iH. Axa,qeMi,{nraft pieexu l.{"! 681
W,ttt,.
!ftffii{:fi;:1:;:
5.49. AQ L a, AB i AC - noxz.ni, QB i QC ri4noni4no ix npo-
exqii, 3 trpffMo(yrHl4x rpuHytHuHin ABQ i ACQ:
BQ'1=ARz'AQ?'BA = {Au" - AA' = 'l:.s' 1r' =.lf€p -r44 = fs = e 1".CQ2=AC'-AQ2;ca = ^tls' -rz" = Jr25 - r44= r,Ei = g (cu).BiAnosiAb:5 i I cm.
5,5O. SO ,L cr, SA - noxtrra, AO - li upoexrlia.SO = 15 cu,AO = x cv, SA= x * 3.3 tAOS (ZO = 9O"):
5.52. MA L a, MB = 13 cira i MC = 15 cru - roxu.ni,AB i AC si4noni4Ho ix nPoenilii.Ocrci,,rr,xn MC . MB, ro AC - 6i,lr,ura npoexqin.Toruy AB = x cM, AC = (r + 4) cu.3 IABM: AM2 = BM2 AB2; AM2 = 132 - x2 = t6g x?.
AM = $6s - b' = "tT6, - % = ^1144 = 12 (cu).Bidnoaidu: l2 cttr.
5.53. cx B, rovrca K j' D ner.<aru y n;rorquai a, M i N -y nrroquHi B.KM L c\, ror.i KM I P. Ocrixrru DN * KM, roDN/p. Tlpote1emo DC I B, ro4i DC - nepnengu-xy.nap i DC = 5 ct't, DN = 7 cM - troxr'ura.
CN - npoenqi" u]4!t""u?ry ua n:roquuv I (s)'
B tcDN: cN =^[DN' a'-UVz -rz =J4s-28=rE4=2J6 (cm).
Bidnoeido: 2rF cu.
5.54, AC L u, AC L 9, AC = 10 cvr. BD - noxlata.Ilporegeuo BK L F, rotl KD - npoexqia BDHa nJrorqrruy S. KD = 24 cw.3 IKBD: BK = AC = 10 cu; BD2 = BK2 i KD2i
5,55, ABCD - ,fiapa!'rerrorpatr, AC i BD - fioro 4iaro-na.ni. 3a yMoBoro AC c a, BD L a, 3a sracrfisicrm4iaroHa.refi trapaJreJrorpaMa AO = OC, ro6ro BO -
5.56, Hexaft B roqnr4 A 4o n;rorqranz cr npone4eno 4rinoxuni AB i AC rax, r4o AB : AC = 5 : 6. flponegemoAO L u, roai OB i OC - npoeHqii noxr.r.nzx. Bpaxo-Byroqfi Te, rqo 6i.nrrra noxr4Jra rrrae 6i.nrtry npoenqiro,
uaemo: OB = 4 clu; OC = 3t[ crt.flosHaqfiMo .r xoetbiqienr nponopqiftnocri, To,qi AB = 5x cm, AC = 6x cw.
Bupaeumo s rAOB i rAOC gonxczHyAO.'3 aAOB: AO2 = AB2 - OBz = ('x)z - 4z : 25x2 - 16.
iii{i:i1:ilti: TEOMETPI9 Sinsnixa O. 9 ra ix. Ara.4eruiqHrfr piBeHb il* 6836A2 ;:::::: FEOMETPIS EincHiHa O. I' ra iu' AxagevrivHuft piBeHb iii:iiili:i;iii
5,59. Hexaft y IABC lC = 90". DC L (ABC), DC = 1 ttn'BC=2tv,AC=3sNr.AB = ^tW + BC' =.,12' .g = G- I = ts (c*).
"s^..- = I AC. Bc =!.r. t= B (amz).22flpone4emo CK L AB, ropi
1 2S..^^ 2.3 6.S -_=:_4B.CK: CK =_:"q=
-=F (AM).
2 AB J13 J133a reopenoro npo rpn neprreHAhnyJrspn, ocxilsxraDC L (ABC), CK L AB, ro DK L AB-
5-63- B qerrrpa ripyra O psniyca OB = 4 cut trpoBeAeMogo ftoro nJron{rrur,r nepnerlAnrcyJrsp OA, BD - Aorr4rr-Ha Ao rcorra, rouy OB I BD, ro,qi AB L BD Ba reope-MOrO npo Tpr{ fiepneH,4rrnyJr,fiprr.
BC = alg cu;,4C = 10 cu.3 aABC:
Au=,171r',-Be= =fioo-48 =Jn=zJrg,
3 rAOB: OA= ^tW - OB' = !62 -16 = ^60
= c (cru).
Bi}noei)a: 6 cn.
$-5-4. Ilepnen,qrarynrpxitTb nroulr,rH- 3anexnicrr Mix napanenusicrpi nepnex4nxynnpHicrxl npcMlrx i nnoulum
Bi1noai}t: 15 cw. D
5.61, Hexafi y aABC AB = 2O cru, BC = 11 cin'AC = 13 cv.V tpnxyrnzrcy rvreumqfi Fcyr JrexrlTr' nporfi Meurrror
cropourr, rovy ZA - uenrurfi.Ilpoae4eno AD L (ABC) i DK I BC, toAi AI< L BCBa reopeMolo npo rpl{ fieprTerrAtlxyJr,spu. DK = 24 cw
3a yMoBoro.
f,Ix i e nonepe4sift sa,qaqi, cfiopucraeMocb norrrr4peHrrM upuiiolrou: 3uaftAeMo flro-rqy rpnr{yrHlr$aABC aa ogHiero $oprrny;roro, a norim a gpyroi @oprrnynra onaiigeuoBHaqeHHfi sucoru AK.
5,75. Ocxim,xu ABCD - HBaApar' ro CD L AD'Ba yMoBoro (ABC) L (ADK), DK c (ADK\' ro*rv CD L DK'3 yADK: DK2 = AD2 + AK2 - 2' + 22 = 8'(cM)'
B rcDK: KC ="ED\ DK" = 'lz'lp = nEz = zJs t"*\.
5.76. 3a yMoBoro SB L AB, SB L BC, AB a BC = B'ro4i aa oaxaHoro neprreuAr'r(yr.fipnoc1i nparroi i nlo-rqunu SB L (ABC).fhorquua (5-419) npoxo.qurb r{epea npamy SB, rouy(s/B) 1(lBc).
AD5,77.HEXAft 1A:aO AABC, NA LAB, MC IAC, NA II MC'
BiAolro, qo srcrr{o oAHa 3 ABox napaJleJrr'Hr{x fipsMrxnepneHAr{nyJrspna 4o rperloi npauoi, ro i gpyra npauanepneHArztcyr.ffpua 4o uei. Touy NA L AC, MC L BC'
Koxna a nprmzx,4N i MC tepneugunyJlflpna uouluui(,4-BC). Toti trroquHa (ANMC), rqo trpoxoArrr, qepe3
npaui,4N L MC, neprl'enqrrlcylaspua nrorquui (ABC) (re'opema 7, uo4y;rs 5).
5.78, Hexaft MC - nepnetAr'rrcyJrsp Ao flJrort\vru ^ABC,MD L AB. Togi aa reoperrom npo rprr [epnel{Aur{yJTrpr4
DC LA.B.Ocxirrxu tnoryuna (MCD) npoxo4zrr vepea npamy MC(MC L (ABC)), ro 3a oor{ar{orl trepneHAnrcyJr,flpnocri u:ro-
quu (MCD\ r @BC).
5.79. Teopema 8, moAy.ns 5.
5.80. Hexaft gsi n.norqunu o i p neperunarorbcfl rro npauifi a i neprer4zxyJr,spui Aon;rorqunn y. To4i vepes 6y4r-axy rouxy npamor a npoBeAeMo fipflMy, nepleHAlxy-rflpHy trJrorrlnHi y. Bi4noxi4llo Ao Br'racrr4Bocri, c@opmylsoeauoi n reoperri g, 4anp,fiMa J'relrcl4Tb i n n.norqrani o, i n n.norquni B, ro6ro a6iraerrcs a npJrMoro a. Snig-cz a Iy,
5,81.3a yMoBoro nJrotr{u}ru o, P i y rolapuo nepneH,qnnyrrsp-ui, PoarrsueMo cx I B. Ilror4rzna y treperuHae qi rrorqNuu norrpfiMrax a i c. f,Irrqo trror{nHv c i B neperunarorr,c,r no flpfi-v'iir b, ay I cr i y I B, ro b I y.A ocni;rsnu npaua b nepneH,4r4nyr'rffpna n.norquni y, To BoHaneprreHAunyrspHa 6y4r-axifi npauifi nrorqnurr y, slca npoxo-AfiT6 qepea rx rouny rreperrrHy. Taxzm uuuou, b I a i b L c.Asaroriqno Moxtna AoBecrx, ryo i a L c.
5.82. Il.norquHu cx, i B nepnen4rHynapni. CD - npnua ne-perfiHy rrJrorqu:n. AB = 25 cm - 4aHufi ni4pisox, A e 9,B e s. AC L CD nigcranr, nig rovnr,A Ao nrrorrlrux o,AC=7cw.BD I CD - ni4craHs ei4 rovxu B Ao n.noulunz 9,BD = 15 cu. 3a reopeuon 8 AC L CB, BD I.4,D. ToaiCB i AD - npoexqii ei4piana -A-B ni4norigno Ha nrrotqu-HHcriB.3 rABC: AB2 = AC2 LBC2;
BC = ^lqnz - de; BC =.JEs" J, = J6% - 49 = G?6 = 24 (crvr).
3 r ADB: AB2 =..AD2 i BD2; AD = J AB' - BD" ;
Ap = J2b' - tb" = .JE2b - 22b = G00 - 20 (cm).BidnociOv: 2O av i 24 av.
5.83. Ba,qa.{a, o6epnena go N! 5.82. flnorqnHN o, i B nepneH-4uHylapHi. CD - npama neperuHy [Jrorrfrzn.
' AB = 25 cm - ganrfi rigBisoH, A e $, B e a.AC L CD - ei4cranr ni,4 rovxu A 4o n;rotluaz o. To4iCB = 24 cm - npoexqir A-B Ha nrorquuy a. BD L CD -ri4craxr, ei4 rounrz B 4o n,rou4unrn 0. AD - 20 crr -npoexqia.4,B Ha nrorquny P. 3a reopeMoro 8 AC L CB,BD LAD.3 IABC: AB2 = AC2 i BC2i
AC = JAa' - BC' = xEb' - 14' = l2b - r4\es. %)- Ge = T (cu).3 yADB: AB2 = AD2 i BD2;
BD=.tqn, An, =J;s, ,0, = [rb-roteb+ror--,li.4E =!669 =
= 5.3 = 15 (cm).Bi1noei1u: 7 crt i 15 cru.
Tecrr 4nn caMoKoHTponK)
9acrrsa 1
(cu).
'12;;v,"
1. f:2. A.
3. A.
7. B.4. B.
5. A.
6. B.
A
D
d
F
D
0
68O i::::: TEOMETPIS Eincxira O. 9. ra ix. Aragevri'{Hvti piBeHb iiiii:iiiiiii;:ii [t:iiitiliii FEOMETPI9 Einsnina O. 9. ra ix. AxaAervri.{xurfi pieexu iii;: 6A7
688 ii::: TEOMETPIS EincxiHa O. 9. ra in. Ara4euirHri piBeHb :iiii:liiifii: :ililt{ili:i TEOMETPIS EincxiHa O. 9. ra ix" Ara4enairxrafi pieexr l{:i: 689
28, aABC - pinHo6e4penwfr, AC - 24 cv - ocHoBa,AB = BC = 20 cru. BS L (ABC), to,qi BS I BK,BS = 8!6 c.u. BK - megiana, orxe, i Brcora.
Y a KBC: fc = ! dC = 12 ctvt;2
BKz = BC2 - KC2 -_ 20' 122 = 4OO - 144 = 256.
3 a SBK: SK' = SB'z + BK2; SK = ,IEA' * nf';
= Jszo. 286 = Jr?6 = 24 (cyt\.
r{acrrxa 3
29. fl.notlurru cr i P - nepneagzxynapni.Hq - "niHia neperuuy trroquH, rovy AH L HQi Bq I HQ (reopema 8).3 rABH: A-82 = AlI2 r BH2.3 aHBQ: BH'= BQ2 * HQ2.Toai AB2 = AI{2 'r BQ' + HQ'.3a ymoeoro AH - a, BQ = b, HQ = c,
ro4i .43 ,E , b" . r'.
30. 3a yuonoro (x LF, KT - risis treperrHy n,'rorluH oi B, rorvry MK I KT i NT L KT aa uacrueictro nep-leHArrnyJr.fi pH14x IIroquH.3 rKMT: KM2 = MTz - KTt'z.
3 aKMN: MN'= KM2 * KN2.MN2 = MT2 - KT2 + MNz.3a yMoBoro MT = a,
KN - b, KT = c,ro4i MN = tE +b, - c,.
31. To.rHuA i B rerxars no pisHi 6oru ni,q rrJrorrIzHu o.AC L a, BD I s. AC = 3O cu, BD = 18 cu.,flxulo ani npaMi nepnen4uHy.napHi 4o 4auoi IrJrorqu-Hn, ro BoHr4 naparemui: AB 1l BD,IhouluHa, fllta rrpoxo.4rrrr, qepes napaJreJrbni npnrrni ACi BD, neperuuae trJrorqvuy c no npauifi CD, a ni4pi-aox AB - r rouqi O.rAOC * aBOD sa ABoMa r{yraMz, 6o ZAOC - IBODfiH Beprrr{aJrbHi, a tAOC i rBOD - npnuorcyrni.IlosHa.rrrvro CO = x cm, ro4i OD = (16 - r) cM.To4i e ro4i6uocri rpuxyrnraxi.e BurrxBae nponopqifi-nicrr, ri4loni4uux cropin:
q
..\
B
25. }JexairAB iAC - lr.i noxuri, npoee4eHi s rovxuA 4otrJrorqr,rHli cr. flpoBe,4eMo AO I oc, -ro4i OB i OC - npoex-qii noxzrux AB i -4C.OB = 27 ctt, OC = 15 cu. Ocxi,rr'xz 6iasura npoexqia niA-noei4ae 6i:muifi uoxr.r.nifi, ro AB > AC.Hexait AB = r cv, ro4i AC - (x 6) cnr.
26, 3 roqxn A Ao tr,aoqr,tut4 o trpoBe.qeHo nepneuAunyJrsp AO i ,4ei noxuli: AB = 4 cttt'
i AC - 8 cv. OB i OC - npoercqii qux noxunux.Ocri.nrxz AB < AC, ro i OB < OC.Hexaii OB = , cM, roAi OC = 7x cv.3 aAOBt AO2' = AB2 _ OB2 = 42 _ x2 = 16._ x2.
690 li::::: rEOMETPIq Eingxina O. 9- ra in. Arag,euisHerfi piBeHb iiiii::-..':iii:i:i: iili:iiil{ii{ TEOMETPI9 EinqxiHa O. 9. ra iu. Ara.qerurirnrafi pieexu iti: 691
neu,qlrxyJrspHltulr nparvri I i CB (aa osuaxoro treprreu-
Asryrgpsocri nparvroi i rrJrorqusr I L (CBM\, s$r\otLBMlILBC).Orrxe, I L (CBM) 3a yMoBn, qo rpaneqii -43CD - npruonyrua, ZB = 94"'
MOAynb 6. Kvrt4 I BIACTAHI v nPocroPl
$6.1. Kyrnynpocropi
6.5. F, r" A.
6.9.8) /ABO.
6.12. B) 10Jd crl.
6.6. r, A.
6.10. A) ZBDM.
6.13. I) 40 c*r.
6.11. B) 12 cn.
6.14. B) LMBC.
6.17. B)
6.20. B.
6.8.
6.1 5. A) flpsmoHyrnraft. 6.16. E)
6.1 8. B) 6-19. B, I.
ffi,ffi
,(,ra nrnanauenna ryrin y npocropi ra poee'asynanu,fl BaAarr e,qiei re*lra neo6xi,q-Ho cHarx osnaqeHH,fi rpr,rronoMerprrruNx (pyuxqifi y npsMoryruoMy rpariyrrrlrftyra ta6.nrzqro ix ana.{eur.Cuny c on rocrporo Kyra B rrpfiMor{}TrroMy rprrrcyrtl4riy Easr,rBaerbcrBiAnoruenn.s fiporr{Jretrcuoro ftarera Ao rinor:enyav:
sina=9: sinB=-b.cc[dociayc on rocrporo Kyra np,fiMonyruoro rprrr{yrur,rna Haar4Baerbcssi,qsorleHHs rrpnJrerJroro ]carera Ao rinorenyeu:
sinz o + cos'o = 1; :::-: = 1961; sin(90'- o) = cos cr; cos(90'- o) = sin cr.cos cr
6.1.f) a = 90". 6.2. E) 90".
AE 2 3 4 a
B
Itymon nixtt Jt uno6iilcHunu npflltumu sasr'rBaerbcfi
rcyr uix npsMr{Mn, axi neperuuarorr,ca i uapa.near'-
ri ganuu nr,rlto6irxnuu [pgMr{M.
I{ymon nix nptLtrtuJttu Ea3ur-
BaeTr,cg nrenmr.rft ia rtyTis,yrBopeur4x nPu nePerrniqr{x np,EMrrx.
m
I aozpannun KA nt'ow uasfiBaetr'cr @irypa, yrBopena
4noua ninn.roulr{HaMll pasoM si cnimuoro fip.rMoro,
u1o ix o6mexcye. Bosa naarasaettca pe6pom 1aoepau''
noao nyma (m)..flrrqo s ^ 9 = m' a r m, a c a, b L m, b c F, roZ(a, b\ * fi:ri\ftrtuia fiyr ABorparruoro Kyra l(a, $.
Idym nitt np,nMon i n,Lour'u-Hon * \e nyr rvri*c rrpgMorora ii npoerc4iero Ha rlro rrJro-
quEy.
CT 0. 30" 45' 60'
srn cr 01
t "E2
t;T
cos fI G2
^,82
I,
tga 0t;T 1 ar6
ctg c .i5 1,6
692 i:.; reouefplg EincHiHa O. 9. ra ix' Ara,4euisHrfi piBeHb iiiil:{iitli :::ititlli:i:{ TEOMETPIf, SinsHixa O. fl. ra iu. Araseruirxrafi pieenu ill 699
6,21.AO L u, AB - roxrJra, OB - npoexqifi roxuror.A.B sa n.nourzuy cx,. To.qi IABO - uryranrzii ryr uirrloxrrJloro i n.norquHoro ctc.
AB = L2 ciu, OB = 6 cm.
cos.lABo=98 = 6 =1.AB122LABO = 6O".
6,22. AO L a, AK - roxuJra, OK - npoerqig noxuloi-4J{ Ha n.norqzny a. IAKO - xyr mixt noxutroto A-I{i n.nouqunoro ct.
AO = 5.tE cu, Ol( = 5 cu.
tE z,qxo = lo LE = Jr,'oK5IAKO = 60".
6.29, AO L u, AB = 4JE c* i AC - noxutti, BO i CO -ix npoexqii. IABO = 60'- ryr ni:+r npauoro-4J3 i o,IACO = 30" - xyr nrix AC i n.norqznoro o.3 IAOB:AO = AB ' sin .IABO = 4rE ' sin 60" =
It=4JB
==6(mr).2
AO63 " AOC: AC = "" = :, 12 (cu).sin30' I
2Bidnoeida: 12 cv.
6.24. AO L a, AB = AC - pinni noxu.ni. Ocxi.nrrcu no-
xnni pieui, ro i ix upoexqii pinni: OB = OC'Togi rABO = IACO Ba rpboMa croponauu. 3 pinnocrirpraxyrnuHin BrrnruBae: IABO = 1ACO.
M
6.26, ABCD - r{BaApar, O -
ro'rna reperrrny fioro 4ia-ronaaeii. SO L (ABC). S1(, Sr, SM, SN - noxz.ni;KO, NO, MO, NO - ix npoeritrii; K, L, M' N - ce-
peAlrHu cropin xna4para.Bonu e ro.rxamrr, B gr[x cropoHu Aortrrtarorbcfl AoBtr[canoro noJra 3 qeHTpoM y rorrqi O.Touy OK = OL = OM : OM r't< pa,4iycr4 BnlrcaHotou tragpar,4-BCD t<ota.
Ocxi,rsHz SO L (ABC), ro nprrMa SO nepnen4r.rxyrflpHa 4o 6y4r,-anoi npnirnoin n.norquni (ABC), qo rpoxo.4srb qepec roqxy O: SO I KO, SO L LO, SO I MO,so r No.a O,KS = aOLS = yOMS = a ONS aa ABoMa r{areraMz (OS - cnilruurl).B pinHocri rpzxyrnrHia BunJrvBae pinnicrr ni4noni4uux t<yria: ZSKO = ISLO == ZSMO = ISNO. To6ro noxra;ri Sf, St, SM i SN saxuresi go nnorlnnu ABCDtr1A OAHar(OBrrM nyTOM
A_____-/'\rl
6,25, O ..._ roqrta fleperuny 4iaroua"reii xsagparaABCD,MO L (ABC).
,{iaroHari xnagpara pisHi i to'moro neperrny gi.nxrr,-
c,s sanni.n: AO -- BO = CO = DO.C Tpltcyrrt:r't<uAOM, BOM, COM i DOM pinni aa 4so-
Ma KareraMu (MO - cni.nrHllfi).3 pienocri rpzxyrnrzria:ZMAO=IMBO=IMCO=IMDO.
6.27. ABCD - trpsMonyrHr.rx, MA I (ABC), MC - no-xura, AC - ii npoenqia. "logi IMCA = 60' - rcyrl.ti*t noxnloto i n;rorqr.rnoro npfl MoltyrHr{Ka.3 aACD:
6,28, Hexafi ABC - piruocroponniii rpr4nyrHrr{, AC c ct,i BC - noxlr.ni, a AD t CD ni4noni4uo ix npoenqii, xyruyrBoprororb noxz.lri a nJTotqt4Horo c{,.
B e cr, BD L o. To,qi 1-BBAD i BCD - xyru, arci
B
L
l
l
IlosHaqzuo cropony aABC - x.tACD nparroHyrnnfi sa yMoBoro i pinno6e4peHzfi,ocHirrnq,A-D = DC stt< npoexqii pieuux noxv.nzx.4Bi BC.
3 IACD:ACz =AD2 t CDz;2CD2 = x"; CO=j;.
3 a BCD: BD = ,lBc, cD' = F:-- = l;' ,Ocni,rsnrz BD = CD, ro r BCD npRnnoxyruufi pinHo6e4penuft B ocnoBoro BC, ro4iIBCD = 45". IABD = aCBD Ba rpboMa cropouaMt4 (a6o ea xarerou i rinoreuyeoro),tBAD = 45".
6.29. AO I a, AB = AC = 3J, cm - noxuri , OB =OC ix npoenqii, IBAC = 6O", IBOC = 9O".IABC pinno6e4penafi s ryrou npz eeptruni60', ro4i ir xyru npu ocHoei BC 4opinHrororr 60",
a a ABC - piruocropoH nik. BC = AR = Syli cn.B IBOC: BO'i OC'z= BC2;2802 = BC2;2802 = L8;BO2 =9; BO - 3 cu (BO > 0).3 t A-BO:
.ao = JAB' - Bo2 ="{s - g - Jg = a (cv).Bi0noei6u: 3 cv.
i
A
694 i:::i: TEOMETPIS Einsxina O- 9. ra ix. Axa4eruivHufi piBeHb iil;ii;iiiii;::i iiiilfi:t:i:lt rEOMETPI9 EinqHiHa O. L ra in. Arasenivxnr pieexu l!:i: 695
r
6.31 . ABC - AaHrrit rpr4ryr:g.vt1, AM L BC - ftoro nrco-
ra, AM =6.,6 cu. BCca,Aes,AMIa.AMLBCaa yMoaoro, rogi MH L BC sa reopeMor) rrpo rpr trep-negArrrcyJrspfi. IAMH = 6O" * .tiHifiuraii rcyr ABorpaH-r{oro KyTa.
3 aAMH: AII = AM.sin60'= 6JE (crr).
Bi1noei}a: I cttr.
6.32, oABC - pieno6e4penurl i np.auoxyrnr,ril:AC = BC, ZACB = 90", AB = 20 cu.AB.(1., C e cx, CM I cr, roAi CM L6ya;-stxifi npruifiu.norqrani o, qo [poxoArrrb qepe3 rovxy M.Ilponegeuo CK L AB, rogi MK L AB sa reopeMororpo rpr,r neprrenAr{Kyrrfipu. ZCKM = 30" - nyr nrirxn.norqnnanrz {ABC) i a.CK - nucora i nre,qiana ar43C.
1
'foai KB==AB= 20:2-10 (cu).2
Bracora CK gir.*rrl, IABC na 4na piano6e4peui npsuorcyrni rpzxyr:uvxn:CK=KB=10cm.
6.33. alBC - pinnocroponnifi , AB c a, C e d, CO L a *nepneEAr.rRyJr.flp Ao rrJrorqr{Ha Ct. ToAi croponz ACi BD * noxu.ni, a niApierdn AO i BO - ix npoercqiirra nJrorqr{Hy d.Ilpore4euo CK LAB - Br^coay ^ABC,CK = 4JB cvr.OK I AB Ba reopeMoro npo rpr{ nepnellArzxyJrffpfi,ZCKO - nyxar.uit xyr uix flJrorqurraMn rABC i a.
s rCKO: sinICKO=co ='J2 =L. 1CKO=3r,.cK 4J3 2
Bi0noei1a:3O'.
6.34. ABCD * r{Ba,4par, AD c a, B e d, BO L a.BA - noxuta, OA - Ii npoexrlia Ha rrJrouluHy o,BA L AD src cropoua KBa,4para, OA LAD Ba reopeMororrpo rprr nepfieH4urcyJr.flpz. Togi ZBAO * ryr ui*cnJrorrlunaMu xsagpara ABCD i a.
- Y aABO: AB = 6 cvt, BO = 3G cM;
sin tBAo = L; =* = +, tBAo = 6o".
BiOnoeida:6O".
6,35. a-4BC - np,nMonyrrrrft (lC = 9O"),AC=BC=6cu.ACca.,Bea.BH La - neptregArrr{yJrxp. To4i BC iAB - no-xrtli, CH i AH - ix npoexqii Ha nJ'IouluHy cL.
IBCH = 45'- Hyr, ni4 arrau rater BC Haxr4Jre-
nraft 4o n.norqrurz. a, ZBAII * ryr rrix ri[oreuy-aoo i n.norqrnnoro o,
6.36. a-4-BC - nprMoxyruklit (ZC = 9O'), AC = BC.AC c s., B e a. BH I cr - uepnenpurcynap, BCi AB * roxr,uri,. CH i AH - ix npoexqii rra rrrro-qrrny d. IBCH = 45" - xyr, axraft yr:noproe xa-ret BC s trroqzuoro u, IBAII - xyr mirx rino-TeHygoro i n.norqranoo o.Y rABC AB = 30 cm, AB2 =AC'I BC2;
AB2 = 2BC2; 2BC2 = 3O2; 2BC2 = 9O0:' BC2 = 450; .
lo3 'BCH BH = BC.sin4b. =|SJZ Y = rb t.*1.
B aABH sinZBAII = BH
= 15
= 1. tBAt{ = Bo..'A8302Bi0noai1o:30', 15 cu.
6.37. Ilnorquu,r o. i B fieperlrHarorbcfi no npamifi /,,Aeu,BeB,AB=25ctl.
AC L I, A9 = 5 cm; BD L l, BD = 8 cn.flporegemo CK LIi KB ll L To.qi IACK - nilt\it-ulrfi xyr ABorpaHrroro ryra uixr trJroulr,rHaMr.r o,
iF. ztcK=60..KCDB -- upfiMoxyrtr4n: KCDB - fiapaJreJrorpaM,6o CK I DB, KB ll CD aa no6yAonorc, ICDB = 9O' ,
rony KCDB * npaqorcyruiax.Orxe,CK=DB=8cm.3 aACK Ba reopeMoro rcocranycie:AK2 = AC2 + CKz - 2AC ' CK . cos 60o;
Ocxi,rsrcu ZBCH = 45', ro
BC =LI\E (or) (BC > 0).
C
@
fr411 =,ls' +8'z -2 s g I = J25+66+-ao = Jag - z (cm).!2
fllorquna (ACK) L I, 6o AC.L I1 KC L l, rouy AK I BK.B .AKB: KB= JAB, - AK" =,1;b" -f = J625-49 = Jb?6 = 24 (cnr).CD=KB=24cm.Bi0noei1u: 24 cyr.
6.38. Il.ror4zuu o i B fleperi,rHarorbcfl no npnrraifi l.Aea,BeB,-4.8=5cru.AC L t, AC = 4^tE cv.; BD I l, CD = 3 cw.flpone4enno CK L I i KB ll I.Toai /ACK = 30' - .irisiftnraft ftyr ABorpauuororyra uinc n.nouqr.rHaMrr o i g. KCDB * [p,fiMor{ylHrrn(o6fpynrynaNus Ar.B. y N! 6.37): KB = CD = 3 cw.I L (AKC), 6o I L AC i I I CK. KB I CD, rottyKB t (ACK), onne, KB L AK.
696 ii:i: FEOMETPI9 Sinsuixa O. 9. ra ix. Aragevi.{Hni piBeHb :;:*:iiti:i;ii ;i{::iiqi:il{ |EOMETPIC SincHiHa O. f,. ra ix. Araseuivnufi pieenu i{!: 697
3 t AKB: AK = 4 at (erurercrxlrft tpuxyrnur).3 aACK 3a reopeMoro xocr.trycie: AK2 --ACz + KC2 2AC'KC cos 30'.Ilosna.rrnuru 4na eplnrnocri ei4pisoH KC auissorc,, orpllMaeMo pisasHua:
6.39, Hexafi-ABC - pirnocroponnifi rpzx;"ruux. AC cct,B e cx. BD I o, rogi 1, i CD - npoeHqii cropiH -4-B
i CB aa rrrorr{rHy o. ToMy ZBAD i ZBCD - xyru,yrropeHi ni4noni4iro cropoHaMz A-B i CD i n.noulra-Horo (x. flo3HaqlMo AJs apyvuocri cropoHy rpfiKyr-surca suissoro a.
'foti s tACD ACz = AD2 + CD2: a2 = 2cD'zi CD = +,12
6.50.
6.52. MN ll AC, rouy MN L BC; KM L (ABC),orx<e, KM L /tlN. MN - ruylcaua si4cranr.BiOnoeilu: !,
{i,fiiEi:iliif:I /: i,l\rf .....!.........: /, ii:41:ii.'..-11!..; ii ii,ili::*:::::::*::_ /_ v?:rir!.'-r.....r.'...'/l i.t $a-'....-..'-'-... ?i I ir;7i1:Jljj.'Ji.,i rrlf ri'...-..r.r.......,:,?_{t:..i-.rtr....'...' ri irt4]'.-.-|:.^...-.-..
6,4O, BO L cx, BA - [oxr.rra, AO - npoercqis noxu.noi-B,4 Ha nrolluny o.ZBAO = 45" - xyr uixt noxuroro BA i nnorqlrHoro o.BA = a. AC c a, IOAC = 45", AC = a.Il.nouluua (BOC) npoxo4rzrr, vepea npauy BO, nepnen-
,qurcyJrspHy n,rollraui ct, roMy (BOC) L s, a BHaqnrL,(BOC) L AO (AO c a). To.qi AO - nepneu4rr(yrflp,a AB i AC - noxa,ri, npone4eai B rotlnu -4 go n.no-rqnuu (BOC), a OB i OC - ix npoexqiiOcxi.nsxn ,4-B = AC sa yMoBoro, ro OB = OC, 6o pinsi noxu,li uarctr pinHi rpoeHqii.Ile osuavae, qo aBOC npauoxyruufi i pinno6egpenufri IBCO - 45" - nyr Ha-
xr,r,ry eigpiaxa BC go nlorquHu ct.
a BOC = t BOA sa ABoMa rcareraMu, azrpcu Br4 rrJrrBae pinnicrp niApisrin BC i BA,ro6ro BC a.Bidnoaida: o:45".
S6.2. BincraHi y npocropi
Bi4crans uint 4eorvra roqr{aMu A i B - AoB}nI.rHa si,4pisrca.4-8.Bi4cranr nig rovrv A 4o npa*roi I goexcuua nepneHAraxyrspa, onyrqenoros roqxu A Ao np,nmoi l.Bigcranr, eig rovxu -4 Ao nJ'rorrlaHr,I o - AoBt{uHa nepneHArtnyJ'rspa, npoBeAeHoroB Toqr{u A 4o n.nou4rznrz cx.
Bigcranr, uir+c napa.ne.nrur.rMu rrpsMoro i n.nolluuoro a6o 4eoma [JrorqrrHaMu - AoB-xcrzna ix cni.nr,soro neptreEAuKylaspa.Bi4crars uix muuo6i**rr{Mr4 flpsMilMff - AoB}KhHa ix cni.nrHoro trepneH,Ithfiy,'Ifi-pa - ni4pisna s xinqauu Ha qux trpsMl4x, nepneHAr4xyJrffpHoro Ao xolxHoi a sux.
6.41 . B. 6.42. B. 6.43- f.
6,44, BC L CD sa yMoBoro, MC I CD 3a reopeMoro fipo rprr flepneHgrxyrspu, roMyMC rxyrcana nigcraHr.Bidnoai1u: B.
698 i:i:j; reOuerplg EincHixa O. 9. ra ix. Axa4enirHui piBeHb iiiillliii;ill:
6.60- .(o4arroea no6ygona: OK L AC, ON L BC, ropi MK1l
oR =;BC 9 cu; oN - raC=s c.v..
MK = ,tmoi\o( =,llr"*r' =
= Jn4+n = Jz% = 15 (cu).
MN=,lMO,'+ON'=rllr'E= Jl69 = 18 (cnn).
Bi0noeida: A,
6.61. Bi,qpiaox KM ue nepernnae rirorluHy o. flpone_pemo KK, L ct i MM r I cr. fllonllrna, fiKa fipoxoAr.rrbvepea npami I{l(ti MML(KKrll MM), uicrrarr ni4pi_sox KM,6o ro.mz K i M sanex<au r1ifi n,rorqr,rni,i neperznae rrJrorqrrHy cr no npalrift KrMr.Hexafi F - cepelr,rHa KM, npore4ervro _f'f, t u.To4i -F, e KrMri KKr, FFr MM1- oi4noni4no ni4_crani nig rovox K, F i M po nroqrrHr,r oc.
?.".:::p:*". (Da.neca F, - cepe4una K,M,. Toli FF,K$MM1.
KK. + MM-tIt= --T-, aei4xz MMr=2ppr-KKt=2.4-l,g =g_ 1,g=6,2(cu).Bi|noei\a: 6,2 cm.
:i:iitiiiti TEOMETPtq Eingxina O. f . ra ix. Axa4eui,rxr,rn pieexu i{:;: 699
A E B3 5 2
6,62. A-BCD - KRaApar, SB -t (/3C), SB = 7 clt'Ocrci;rutu SB I (/-BC)' to SB -L BD, orxce, nigcranrei4 rourcu S ao np,nuoi tsD 4opinuroe 7 cn.
,{iaroira.ni rcnaApata neperrrHarorbcs nig lpauuu xy-rou: BO I /C. Toai SO L AC 3a reopeMoro npo rpr'I
neprreH4lrnyJTspn, SO - rigctanr ni,q touxu S 4o
,qiaronari AC,
BD=ADJ'=8J'no =!no = +Jz
so =,[sn\ no';r--- . --.
so = J7' *(+Ji)' = J4s.n= Jsr = s {c'a).
Bi1noai0u:7crvri9cm.
6.63. Ilponegemo MO L (ABC) | OK L CD.'fogi MK L CD sa reopeMoro npo rpll nep-
rrerr.414r{yJr,'IplI.
MK * siAcranr ni4 rourra M 4o ctopoura HBaApara'
Ocri.rrrcu pinni noxra.ni, nporegeni a o4rdei rovnu, rvra-
rorr pinni npoerqii, a 3a yMoBoro nigcrani ni4 rouxra
M ,4o nomnoi croponnr r{Bagpara pinni, r:o i ix npoex-qii pinni, ro6to O - rleurp Brrucauoro fi onr.rcasoro
rraBRoJro KBai4paTa ICoJra.
AD=7ocrvr, OK =r=+={=r f"*1.223 yMOK: MKz = MOz * OK2;
6.65. Bi,qcranr niA tovrcv M Ao ecix Beprrrr{rr :crra/'palaABCD oAHarcoBa (M q (ABC))'
MA=MB=MC=MD=5cw.Ilponegemo MO L (ABC), ro,qi O - 4elrrp r{orra, oruca-Horo EaBrdoJro ntagpara ABCD, 6o OA = OB = OC = OD
ar npoenqii pinuux noxu.nax.1
V a MOC: OC = - AC = 2,5 cwr.2
MC2 = MO2 + OC2; MO = .{Me - OC, i A
Mo = ,F - z,s" = Jr,l '? 5 = z, rJF (cm).
Bilnoeida:2,5n8 cM.
6.66. AABC - rp,nMonyrur.ria, ZC = 90", AB = !2 cwt. S e (ABC\.Ilpooe4emo SO L (ABC) - nigcranr, oi4 rovrra S go naorqrznn (/BC).SA = SB = SC = 10 cn - nigcrani ri4 rorxz S go nepurr.tr a-A-BC,Togi, OA = OB = OC ax npoexqii pirHux noxra.nnx narrJ'rotr{t4rry (ABC}. Togi O - qenrp ftoJra, o[rrcanonoHaBr(oJIo \ABC, - cepeAr,rHa rinorenyara AB.
6.67. IABC - rpaMonyrrrqit, lC = 90', AC = 3 cn, BC = 4 cm.S e (ABC). Hexafi SO L (ABC) - nigcranr ni4 rovnra S 4o nroiqtrur.r (ABC). Ocxins-rcr S,4 = SB : SC = 6,5 cu Ba yMoBoro, ro OA = OB = OC sx upoerqii piarux no-xtrJurx Ha nrorqr,ruy (zlBC).To,4i O - qenrp KoJra, ofificanoro HaBHoro aABC,O - cepe4r,raa rinorenyor,r AB. AB = 5 cu (a.A-BC -cnrnercrxr,rft),
oa=L*en=2.b cu.2
B aSOB: SB2 = SO, + OB2; SO = nlSA, - Onr;
so = J6,E' - Zs, = il65+r,11(65 - r51 = Jt 4 = B 2 = 6Bidnoaido: 6 cttr.
6168. V rcragpari MNPK O - roqr(a neperzny 4iarona-nefr. AO -E'(MNP\, nO = 2',12 crvr, MN = 4 cn.AK.= AM = Al{ -- AP - nigcrani ni4 rovxu A 4oBepruuu r(BaApara. Bonr,r pi*ni ar noxu.ni, sxrawr ai4-nonigarotr pierii npoexqii. KO = MO = NO = PO sxnoJroBr,rHr{ pierrax giarona.tefi rna4para.
6.69, .(aao IABC, AC = L0 crra, AB = 17 crrn, BC = 21 cu.V rpzxyrnraxy HaBnporrr 6i.nr,nroi cropoHr.r ,rer$rrrb 6irrl tnuir xyr. Onne, ZA * naft-6i.nrruvft. DA L (ABC). DA = 15 cu.flpone4emo AK I BC - Brrcory rABC, roni aareopeMoro upo rpra neplerrArxyJrffpu DK L BC.DK - ei4craHr rig rovrcu D 4o cloponra BC.Br,rana.rzmo AoBlrcr,rny DI{ o npauoxyrHoro rprr-xytlelzxa ADK (DA L (ABC) , rowy DA L AK) . !,naqboro EorrepeAnro euaftgemo :4J(. s OopMyrr.r Ars
cwr (ax 4iarona;rr rcna,4Pata).
cM.
ffiffi*f;$ffif,ffi
(cu).
LO+L7 +21_-rL,2 --'
NJIOII{I TPIIKYTHfiKA:1 2S*.
S^"". =: AK CD: AK2cBflrorqy Br,r3HaquMo aa Qoprry,rorc fepona: p =
7OO i::::: rEOMETPI9 EincHiHa O. 9. ra ix' AxagewrivHrf piBeHb:i:::i':::i:i:C: :iiil*it:ii*i TEOMETPI9 EinqHixa O. 9. ra ix. Axa.qeuivmnfi pieenu l:ifi zll
lB - lr.afi'mesllnufi, ocxi.nr,xu rerrtI'rrb HaBnpor[uesmor croponra AC.MB L (ABC\. Ilpoee4euo BK L AC - Br{cory*ABC, togr Ba reopeMoro trpo rprr nepneuAur{y-napu MK I AC, MK - ei,4craHr ei4 rovnu Mgo uaiimennoi cropoHu AC.3a yuoeoro MK = 15 cv'. MB ni4craun aigro'rxu M Ao nrrorqrrHtr (ABC). Suaftaevo MBB npsMoltyrHoro rputtyruutta MBK.
"logi ABCD - trpsMoxyrHun.
AC2 -AB2.t BC2 =AB2 i AD2; AC = JrO\rt' = fqll+44L = J841 = 29 (cm).
Bi}noei1u: 29 cw.
6.72. fleprenguxyrnpui trroilIlrHr o i B nepetttuarorr,cs tro np.sMifi c,
M@s,MeP.Ilpose4emo MB L o., MD L 9, MC L c.
BC .'._ npoexqis MC nannonluny ot,, roMy 3a reopeMolo
rrpo rprr neprreHAnftyJrfipu BC L c, a sa s.nacrunicrlotrepner{Aur{yrrflpHux rtJlorrlnu (S5.4) BC I o.Octti:rcxn MD L B, ro ea e:racrznicrro nepnenguxylap-nocri npamoii i.n.norquHr.r (55.2) MD ll BC, a asa.rrzrr,,MD i BC rexarb e oAHifi naouzui.Toai MBCD - flpflMonyr]rux.MC2=MB2*BC2=MB2*MD';
7O2 l::::: ;EOMETPIS Einnxixa O. 9. ra ix. Axa4ervrivHl4i piBeHb ii:ii::ii:i:i:::i:i i
6.73. ABCD - nBarpar, M e (ABC), MA = MB == MC = MD. Ilo6y4yemo MO )- (ABCJ.IAOM = aBOM = aCOM = aDOM sacninr;eumuIcareroM OM i rinorenyaoro.Toti AO = BO = CO = DO. O - rorrfta neperrHy4iaroHa"re* HBa4para.flronlraHrz (AMC) i (BDM) neperwHar)rbcn tro Anpamifi MO, MO L (ABC\ sa no6ygonoro, rogiMOLAOiMOIDO.Ilrorqzna (.48C) neperzHae nr'roqr4 Hr.r (AMC) i (BDM) no rpnMux AO i DO.AO L DO arc npsui, rqo uicrarr 4iarouali ftBa"qpara. Togi (AMC) L (BDM) Ba oBHa-rreHHsM trepleHA[KyJaEpHrrx nJlorqrrH.
6,74.3 roqnz O neperrruy giarona.nefi npflMoxyrHlrxa ABCD nponegeuo treprrenAuxy-.nap, Biar,lremo Ha HboMy 4osiar,ny lowy M i s'e4Haervro ii s repurHau[ npeMo-KyTHrKa.Ocmirsxn aiarosaai npfiMonyrHuKa pisui i rovnoro neperuHy 4irRrscn naenir,AO = BO - CO = DO. "Iogi rAOM = rBOM = tCOM = aDOM Ba ABoMa KareraMr4(rpzxyruznu npmronyrni, 6o MO L (ABC), or*.e, MO I AC, MO L BD),3 pinnocri rpuryruzxin nunnwnae:. AM - BM = CM = DM, ro6ro roqrca M pinuonig-AaJIeHa B1A BeplfirlH trpsMOKyTHlria.
6.75. ,{aHo ni4picox KM, au:aft He neperuHae nJaon{r4Hy
cx,. qepes Hinqi K i M ra cepe4zHy F ni4piana I{M: trpoBeJrr.r nepfienAraxy!'rsp[ Ha trroqqxy a: KKp MMr
i II, (nia.ctaHi ni4 rovox ,4o nrou{r4Hr4 cx). Ocxi,rrxuKKr L a, MM1 L cr i FFr -L u, ro KK,ll MM, I FFr.Ilpoee4euo uepea npaui KKri MMr nrorquxy, eonarreperr.rHae trroqzHy o no rpauiii K,M,.[ei rouxu npauoi KM Hare]rcarb (K$M), rot'ty KM e (K$M\. F, ua.neuczrt nro-l4lnai (K.KM) i o, romy Fr-F KrMr. Tanuu uuHom, rpavri KM, KrMr KKr FFrMM, temarr e o4nifi n,roruuui.3a reopemoro @a.neca I, cepeArrna KrMr. To4i FF, cepeAus :rinis rpanerlii
K,K M M ,. FF, = KK' t=M *' . lt\o i;i rpe6a 6y,ro .4onecrz.rr'2
6.76. Hexaii rovxa S, trIo Jaerfirrrb rrooa trJorr{[Horo trpflMo]ryrHoro rpraxyruuNa /BC(lC = 9O'), BHaxoArlrbcs na nigcraui 10 cu ei4 nontnoi fioto eepruzuu:SA=SB=SC=10cm.Ilpone4e*ro SO L (ABC), rogi AO = BO = CO nn npoen-qii pinnux noxzJrr{x, ro6ro ro.rxa O -
qeHrp onucaHo-ro HaBrtoro rpzxyrHzxa-43C ro.na, V npamoxyrHouyrpr{nyrHr.rxy eiH ler*curs ua cepe4nui rinorenyaa.
SO - mynaua sigcrans si4 rovxrz S 4o llorqzHu (-4BC).
6.77. ABCD pom6. M 4 (ABC).Ilponesemo MO L (ABC) i MK L CD. MK BiA-craHr nig roqxv M Ao cropoHr CD. MK = 8 cm. OK L CD Ba reopeMoro npo rpzn epneHAr4KyJspu.
:iiii;:ll{l:ll TEOMETPIC Eingnisa O. 9. ra ix- Ara4ervrirHr4fi piBeHb iii{i 7Og
M
BK anaft4erro B (DopMy,au n.irorqi rpHrcyrHuHa: S*" = 1o" BK; BK =2
MB = ^[MK" - BL: = jrb'z -1r'z = tEE -\44 - rfal= s (cu).Bidnoaida: 9 an.
2S*"AA
6,71. Il.norqunu o i 0 neptreHAuxyaspni i neperunarors-ca no npauifi c. A e a, A e F.Ilponegemo AD L a, AB L9, AC L c'BC - npoexqis AC Ha n,rouluny B, touy 3a reopeMor)
npo rprr neptrenAur(yrJrpu BC L c, a aa o.nacrunicrronepneuArny;r.ffpHilx nJrorqrn (S5.4) BC I g' Ocxial-xtt AD L cr, ro ga e,Tacrusictro nepnen4unyaapHocrinpauoi i rrrlorqr{Hrr (55.2) AD 1l BC, a a+taqurs, AD i
D CI
a"
BB
BC .nemarr n o,4Hiit nJrorqrrHl
D B
cM
C
Ba
l1;,Ii.lr,i;o.':.:.i:r.:,:
,1r r-nski-r^.rq.ir-.-..
'Jlisk.:t !t.c.r.:.
Bi,4cranr OM s\F,ro,txu M uoxna snafirz 3 np.fiMoxyrHoro rplrrytuuxa MOK'Ilonepe4nro exaftgeuo AoBrnl4Hy ni4piaxa oK. lde uoxna apo6urv, BI,IROprcroBy-
rorru nnouly a COD.
,{iarora.ni pou6a nepneugrrcylapni, ro'{rca O gi.nunr ix naenir'
3 tcoD: oc =f,ac= f; = a 1"*1;
oo--!no-r]=a <"*t.
cD = Joc' * oD' = J8' + 6' = Jroo =.ro (cm).
s^"oo--!oc oo=t 6.8=24 (crvr'z). A
s*- = tco. oK; aK ='* =# =4,8 (cu)'
3 tMOK: MK2 = MO2 * OKz;
Mo=.tMK'-oK' = l7 4,8'= Ji8-4fX8+4s) = JtTLr,s --
= Co, 16 = 6,4 (cm).
BiinoeiOv: 6,4 cw.
6.78.. ABCD - pinno6iuua rpaneqia. AB = CD. P *"o = 48 cw' 1A = 6O"'
To.rra S He Jre)fir4r6 y n;rorqrzni rpaneqii. flporegemo so I (ABC). OCni.nrHra toqxaS piBnosiAAalena aig cropin rpaneqii, ro i tovrca O pisuoniA,qarena ri4 nzx, 6o
pioni noxu.ni uarort pinni upoerqii. Tomy o - qeHrp noJra, Bnr{caHoro n rpaneqiro.
To.*ca O - cepeAfiHa ni,qpioxa MN, qo e'eAnye cepeArrHr'I ocnon rpaneqii'MN LAD, MN 11 BK, MN = BK, pe BK - nzcora rpaneqii.SM - lryxana eigctaHr ui4 touxu S go cropoxu rpaneqii, 6o SM L AD sa reo-peMorc npo Tpr{ ilepfieHArrrcynnpu. ocxi.nr,ria n rpaneqiro Mo]r{Ha Brrr4caTI,I I{OJIO, TO
cyMa AoB)r(uE ii ocnos gopinnroe cy*ri 6iunux ctopin: .
pAB + CD = BC + A-D = -T = 24 (cwl,
AB=L.24=12 c*r.2
3 ,ABK: BK = AB.sin 6o' = n g= 6.,6 (cu)'2
NM = BK = 6V'B cru; OU =\Nttt = 3175 civr.2A3 aSOM: SM2 = SO2 + OM2i
6,79. y IABC AB = 26 em, BC = 28 cw, AC = 2O ctt.M e (ABC).Ilpone4emo MO I (ABC), MO - nia-craHr nig toqnra O Ao flJrorqurrt4 Tprtryrnr{na.flo6y4yeuo MK L AB, ML L BC, MF I AC, MK,ML, MF - ni4ctani oig ro.rrcra M 4o ctopin tpu-rcyrnuna. OK, OL, OF * ix npoerqii. 3a yuonoroMK = ML= MF = 17 cu. ToaiOK -- OL= OF, rouxaO - pa,4iyc r{oJra, Brrr,rcaHoro n rpr'rxyrnux -ABC.
,[onnruny ri4pisrca MO zuaftgewo s tMOL'
Ol snafigemo s (popMyrl4 . = { , g" r - pagiyc Brucanoro KoJIa, p - ninnepu-
pMerp rpnl{yTHffKa.
7o4 iii: fEoMETPlg SinqHixa o. 9. ra ix' Axa4ervrirHufi piBeHb ii::.:.r:{;ii:ii
Ba rpopny;roro fepona: o - 26 + 29 + 30 = 42 (cn\;
6.8O. tABC - pinnocropouNlit, P*"" = 36vE cm. S e (ABC). flpore4euo SO L\ABC\,SN l/3, SK I BC, SM IAC. To.4i SO - ni4cranr ri4 rovrcr S no nnon\uau (ABC).SN, S,M i 51{ * nigcrairi ni4 rourcu S 4o cropin IABC. ON, OM i OK - ix npoex-qii. Ba ynnororo SN = SM = SK = 10 cn. ToAi ON = OM = OK src npoexqii piniriaxnoxr4Jrux. O - qeurp Bnrrcanoro s aABC no,na, ON, OK i OM - fioro paAiyctr.
AB=BC=46=Pwc - 36'FB = g =12V3 (cu).
^ AB"J8 L44.8 t;S*. = '*Z*" = 108J8 (cur). A
oK =21n.*:= ,l1ot--Jt
= 6 (crvr).P*"" 36J3B a SOK SK2 = SO2 -f OK2;
so= Jsr, -oK, =Jro, -6, =Jr00-g6 = Jca = g (cu).Bi)noaidu: I cm.
6.81 . ;ABC - pinuocroponuifi, Smc = 27JB cm,.Toqxa S piuHori44a.neira nig nepurran a-43C.AS=BS=CS=10mr.Hexafi SO L (ABC), rosi OA, OB i OC - piniri npo- A
'ercqii pinulrx noxr,rJrux, O - qerrrp KoJra, on[cauorosaelcoao a A-BC.8nafi4euo cropony rpr4nyrnr,rra a @opmy,ru fioro n;rorqi:
6,85. flprui m i n unno6ixtni; n c a, m Lo', A € m.
Hexafi O - roqxa neperrrny npamoi m e nnorqlrnoro tx. To4iAO - ni4crans ei4 rovnrz -4 4o n.norquurz o. OA = 6 clt.Ilpone4emo OK L n, OK - ei4craur mirx mumo6ixnrauutrpsMuMrt m i n (ix cni.nsszfi nepneH4rzxylcp).3a reopemoro npo rpu neptreHArrnyr.fipu AI{ L n, A-K BiA-
craHr eig ro.{xz -4 4o lpamoi n, AK = 1O cxr.
3 yAKO: AI{2 = AO2 r OK2;
oK =,tqx' - do" = v[0' -d - .,[00 -36 =.,62 = s 1cu).Bidnoaida: 8 cn.
npau npoertqiroBaHH,tr trepfieHAuxy napuuit Ao nroilIuHtr npoeHqii'Teopeua. Illorqa oproroualrHoi npoexqii ,qoninsuoro MHOTOHyTHIIICa Ha ITJIOIqIIHy
4Opinnroe 4o6yrxy n,rorqi cauoro MHoronyTHuKa Ha ROCIIHyC Hyra uirx trJIoIqIIHOIO
@irypn i n.nourunoo nPoexuii'S'=S.costp.
6.86. A. 6.89. r. 6.91. B, tr. 6.93. B. 6.95. B.
6.87. B. 6.90. F. 6.92. A. 6.94. .U.
S^*" - S^*. cos cr; s-r" = &le
A) swr = #h = Tr, = + = r2Js (crur,):
E) s,,,* - t:," -18;2,r2y[2 (cvzr:
cos4D" \1218
= ra:1 = 36 (cu2).cos 6Oo 2
84cos0 = - = _:
10 5
6.98.
6.97. A.
S.or" = S"o"" cos o;
"oro=ffi
B) cosa = I = 1,' 15 3
93R) cosn=l=:.'21 7
_61ll cos0= = :'122'' 1t t
lll cosa = =183
B) Swr =
A)
fifi,ffi
6.1OO. .41{ L (ABC), AD I DC, rouy KD -L DC (aa TTII).
S.-. = S"no, . cos 45'; {-- = "flTr".
=V=*f =B2li @nt).
6.101. S€'* = Sor"r'cos60o. S*"o- AB . CK = 15. 12 - 180 (cu'?).
1S*,,, =180 t=90 (cv'?;.
Bidnoeida: B.
6,102. aABC i IA.BD - pirnocroponni.
AB ' AD - BC l',6 cm.flpoee4euo CK L A-8, T oai DK L AB ea reopeMoronpo rprr nepflneH,4rrxyJT,flpu. ICKD = 6O" Hyr mi:HnJrorIZHaMu rprHyruzxin -4-BC i ABD.PosrrsneMo rBCK. KC - Bucora, a BHaqurb,i ue4iaua tABC; IKBC = 6O'.
6.108. A e cr, B e a, C e AB, AC : BC = 3 : 5.Ilpouegeuo CC I ll BBr Togi AB, - npoexqiaei4piexa.43 Ha trJrorqury ct". Cre ABrOcxi.rrxia CCr ll BBr ro aACC. o al3B,.AC : AB = AC, : ABri 3 : (3 + 5) = AC, : 48:'
AC. = 3' 48
= 18 (cut.'8
Bi}noeida: 18 cpr.
6. 1 09, Poarr nHeMo AJts cpylnocri oproroHa,rb-Hy npoexqiro 4aroi rpaneqii ABCD ua nao-rr$4rry o, gxa micrurr ocnorylS i napa;re.nr-Ha ocnosi CD.Ilporegeno DD, L a i CC\ L a. ABCTD, -oproroHaJrbua npoexqia rpane4ii ABCD uanrorllrHy d.Ocni.rrxr,r DC ll a, ro DC ll DtCr DIDCC. -rrpffMonyrnrrx, Dpl= DC = 3 c1a.Ilpone4errno DE L AB -- Br{cory rpaneqiiABCD. DP LAB o,a reopeMoro rpo rpu nep-rreHArarcyr'rffpfi. Togi IDED. - ryr mixc n,no-ulrruoro rpafferlii i nrorqznorc lpoexrlii.Ana.noriqno CF LAB, Cl IAB.
Ba r{areroM (SO - cni;rrnxfi) i uporu.ne*car{M roc'lrpr'rM nyroM.
Snaqrrr, oM = oN = oP = oK, ro6ro 'ro.rxa o pinnoaig4a;rena ni4 crnpirr tlorupu-
xyrwxa ABCD. O - rlenrp KoJra, Btrr{cauoro s ABCD.
6,1O7. Ba yrroBoro (/3S) L (ABC\. SO - sncoranipaui4u. SO L (ABC\, rorray SO c (ABS).flpone4ervro SM LAC i SK L BC, roaioM LACi OK I BC aa reodeMoro npo rpr{ nepnenAil-KyrI.fipu.ISMO = ISKO - pinni 4aorparHi ny'tu, y"reo-
peui rpannur 3 ocnoBoro. ISMO = a SKO sa
KareroM (SO - cni.xlnr'rft) i nporunerxuramrocrpr{M xytort. Bnigclr OM = OK,tMOC = aKOC sa KareroM i rinoreuysoro.To,qi OC - 6icetrpuca rcYra C..
B a DED,: .o". onn, = lE 3"8 "6 rr""o. Jt .DE= i=:. IDED,=t
4
20 1
40 2'
6.110. 3a yMoBoroS ll cr. AB nCD = M,CD n o =N. ICMB = 4b".rlepea npaui AB i CD, ulo neperrruarorr,cfl, [poxoArrrb rrj'rorquna (ABC\, ax,a ne-perrrHae nrrorqr{Hy cr no npnuift PK, PK ll AB. IMNK = 45" strt ni4noeignnfi xyryCMB lrpw napaJrerrbHr4x npsMrzx PK i AB i ci,a:r.ifr CD.flporegeuo MO L cr. To4i NO - npoercrlia nparroi CD Ha nJrorlzny cx,.
IMNO = 30' - xyr uirrc npauoro CD i n;roulunoro o,Ilpone4euo MF L PK, OF L PK sa reopeMoro npo rpt4 neprrenAzlcyJrffpu. To4i xyrMFO -;risiftnzji rcyr ABorpaHHoro Kyra, yrBopeuoro firou]rrHaMrr (ABC) i a,flosHaqfiMo MF = a. Ocni.nuru s d MNF IMNF = 45", ZMFN = gO.,
ro i INMF = 45", ro6ro a MNF - pirHo6e4penzfia ocgoBoro NM: NF = MF = a.
NM = ,[MF\ Nr' = ,17 + o' = oJz.
SaNMo: Mo=Luu=ofr2 2
sH HaTeT,
rrlo Jre)firrrb HaBnporrz Hyra 30o.
3 t MOF: sinlMFO = MO - a'r2 . - -'',tE'MF= 2'o= 2'
ZMFO = 45". II{o fi rpe6a 6y,ro 4oeecrz.
7oa iii: TEOMETPIS EincxiHa O. 9. ra ix. Ara.qeuivHxfi piBeHb :iilii'.::.fi::r iiili:ltiitil TEOMETPIC EincHixa O. 9. ra ix. Axa4euivnrafi piaexu;ii:fi 7Og
ny npoexqiro Aairoi rpaneqii ABCD sa nno'rrluHy ct, ,srta uicturs ocuony /-B i napa,'re.irr-
na ocnosi CD.flpore4euo DDr L cr i CCr L s- ABCTDT -oproroHaJrbHa npoenqir rpaner{i ABCD saIIJIOIIII4HY d.Ocrcirr,xu DC l1 a, ro DC ll DtCr DtDCCt -trp.fiMonyru[n, DrCr= D( = ]' sNt.
Ilpoae.4evo DE I AB Br4cory tpaneqiiABCD. DP LAB 3a reopeMoro npo rpr4 rep-neHAnHyJlflpr4. 'IoIl IDEDt - Hyr mint n.no-
rrlrHolo rpanerlii i n,ror4r'ruoro npoexuii.Asa.noriqno CF LAB, Cf IA-8.
Ba n.nacrneicrro piaHo6ivHoi rpaneqii AE - BF =
D. t: 3,13 .23aDED.: DE.--t.-: -- =b (cM).
' cos 3U" J3
ABTDC -- t'o
.s --- ='*':- .DE="'-1 .6=36 (cM':).-MtD 2 -- 2
6.112. IABC i aABD pinHo6e,4peni, ,4-B -cri,tr,sa ocHoBa.flpose4ervro CK L AB, roai CK - Brrcora
i ue.qiasa pieso6e4peuoro \ABC, rowty K -cepe4nua A,B.
V tABD KD - megiaua, a auaqurs, i snco-
ra: DK LAB. zCKD = a nYr uinr n.no-
rrtrrHaMv (ABC) L (ABD).Onyctuuo nepneuAurdy!'r,flp CO na n,roqrzny(ABD). "AOB oproroHaJlbHa npoexqintABC sa nrourrauY (,4-BD).
OcHirtxz CO L (ABD), (KCD\ L (ABD\' CO c1
3 IAKC: AK -:AB=24:2=12 (cv).2
1Bi1noai1a: o. = arccos-
30s: - cM,
7
6.113. IABC i rABD - pieuo6e4peni, AB -cnimsa ocHoBa. AIfpone4euo CK I AB, roxi CK - Bhcorai megiaxa pinno6e4penoro IABC, rouy K -cepe4uua AB.y
^ABD KB - me4iaHa, a sHaqrrrb, i nHcora:KD - AB. ,zCKD = 0( - Kyr vix n,rorqznauz(ABC) i (ABD).flpone4euo CO L(ABD), IAOB oproroHarb-na rpoercqis aABC aa ntot4uny ABD.Ocxi.nsrcu CO L (ABD), (KCD) L (ABD\, CO c (KCD'), O
19. MO I a, MA i MB - noxra.ni, OA i OB - ixrpoercqii. MO = 4 cw, ZMAO = 30", ZMBO = 45",IAOB = 150".
BaAoM: lo= Mo =!.3 =126=+,8 r"*r.te30" JB 3
B a BOM: OB = OM = 4 cu (npsuonyrHzft piano-6egpenrafi rpuxyrnzx).3 tAOB 3a reopeMoro rcocnHycin:
AB AO'z + BO2 - 2AO . BO. cos 150'
-
V48 + 16 + 2 413.4. cos30" =
(cnn).
Bidnoai|u: 4J7. "*.
2O. fl.norqznr,r o i B neperzxa"rorr,cs no npamiii c. flpo-ne4emo -4D L a, AB L I, AD i AB - ni4ironi4no ni4-crani ni4 roqnu A 4o nnorquir cr i B.Il.noiqr.ria (/.BD) npoxo4lrrr, qepeo lpav.i AB i AD,rotly (ABD) I a i (ABD) I B, ro4i (ABD) L c.1"DCB + 6O' - xyr uirx n.norqruarru cx i B.dABC = aADC salg"anerorr i rinorenyeoro (-4C - cni;rl-na rinorenyaa, AD = AB = 10 cel).VTABCZC=60.:2=3O'.AC = 2AB = 20 cu (-4-B - xarer nporz xyra 30o).Bidnoaidu: 2O cwr.
21. fllorqunu cl i B neperuuarorr,c,ff flo npruift c. flpo-ne4emo 8.4 I u, BD L P, BAi BD nignoei4no ei4cranini4 rovrcrz B 4o n.rrorqun cr i B.ffurorquna (ABD) npoxo4urb rrepea npaui 243 i BD,rouy (,43D) L a i (ABD) -L B, ro4i (ABD) L c.IACD = 60" - rcyr uirx naorqunamu cr i p.
d BAC = t BDC aa KaaeroM i rinorenyaoro (BC == 16 cm - cni.nsua riuorenyaa, BA = BD sa yMoeoo).
22. a LB,Ae o, B e B, o. F=c.AA, I c, AA, = 3 cur; BB, L c, BB, = | sNr.
44 =.,,[T "*.AA, e a, AA, L c, rouy,4-4, J' B,orxte, AA, I ArB.3 aABrB:
ArB = JTq * BBi = "41-
16 =.@ = sJ5 (cm).
Ga+s2r6 \=[a-aJV
3a2
2
fi{iii{il:ii TEOMETPI9 EincHixa O. 9. ra in. Axa4euivxr,ri piaexu ii{:;i 719
3 .AA,B Au = rtr4 - 4n', =,tl',; n = t6;n = 16o = o (cru).
Bi4noai}u: 6 cv.
23. fl;rorqnnz o i B nepernrarorbc.a no npnulifi c. A eflpoee4elno AD LaiAB LF.Il;rorqrzua (/3D) npoxo4urr uepea npsui-4-D i-4-8.
Touy (,4-BD) I c.tABC - aADC sa KareroM i rinoreiyaoio (AC -cni.nrsa rinoreHYea, AB: AD = a\.IACB : IACD = 120" : 2 = 60".
Aa = a :sin 60" - a'-2 - 2a'13 .JaB
2a,liRidnoeida:: cM.3
24. (ABC) f 6BC), rABC = r KBC - pinHocropouni.AD L BC, KD I BC - Brtcora (ronn x me4iaHir) aABa i tKBC' /ADK - nisifi-nuii nyr ABorpaHnoro xyta e pe6pou BC, IADK = 9O".
.A,I{ - pe6po ABorparrHoro ryra uixc nrrorrlrrraMu (AKC) i (ABK)'flponegeno BM LAI{, rotii MC LAI{,6o IABK i *ACK - pieHo6egpeai a ocHo-
r'ow AK. IBMC - ryr uil+c nrroilInHaMlr (AKC) i (ABC).
flosHaqumo cropoty piaHocropoHnr,olo rpl'rnyrsuna a'
B ^ACD: co = ! sc -- o., zACD = 60";22
^ti BAD - AC.sin60" - "v" .
2
DK = AD= of
r"*1.
3 IADK: AKz = ADz * DK2i
26. ABCD - r{Ba,qpar ei croponoro 6 cu. F e (ABC).AF-BF=CF=DF=7cM.N - cepeglrna FC, M - cepe4zua-4-B.IlossaqnuoP - cepe4zxy MF i K cepepnay CD.Poar.n.sHemo qortrpuryrHzn MPNK.
28. MO L (ABC), MO = 3 cu - ni4cranr ni4 rouxr M go ntot4uuu (ABC).Ocrci.nr,rcra ro.rxa M pinnoni4garena eig cropiH IABC, ro O - 4enrp noJra, Bnuca-Horo n a-4-BC, - Toqxa neperr{Hy ue4ian (nucor).Hexaft AD - ue4iaua i grcora IABC, rogi MD L BC aa reopeMoro trpo rpr.r trep-neHArrnyJr.rrprr. MD - ri4cra:ur ni4 rovru M go cropouz BC.
MD = 2J3 cv.B t MOD: MDz = MO2 + ODzi OD = JMD" _ MO, .
i cropoHn npaBl,rJrr,Horo rpr{}cyrHuxa a 2rJ3, roilBC = 2. rE "6 = g (cu).Bi)noei0u: 6 c*r.
roa2 loaz+ -a'16 16
;r@!@44
CI
c
D
A
b0
6a2-16
714 ::::::: rEOMETPI9 Sincxixa O. 9. ra ix. Ara,qelurirHufi piBeHb liiiii:;iiliiii [itilitilli{ TEOMETPI9 Finsnina O. L ra ia. Axaseuivxun pieenu l{:i Z1S
Ilponegeuo SO L(ABC\. O * tovrca neperr'rny meAian (uucor) aABC' SO - niAcraursi4 rourcz S go rpani r4SC. S
OB - pa4iyc onrrcauoro r(oJra.
3 oSOB: SBz = SO2 l- OB2:
MOAy,nb 7. ySArAIbHEHH9 I C]4CTEMATt4SALllg B]4BqEHO|'O
S 7.1 . Ocnoeni Sirypn reoMerpir'ra ikne posniu4eHHe y npocropa
o"=#'
7.1.A.7.2.t.
7.3.
7.4.8.
7.5.4.
7.6.8.
7.7.8.
7.8.r.
7.9.r.
7.10. B.
7.11. r.
7.12. A.
7.13. A.
7,14. L.
7.15. E.
7.16. f.7.17.8.so = JsB' - oB' =
BiAnosiAb:
30. MO I tr - nepner,qlarcyJr,Ep. flosua'ifiMo fioro gonrxi4ny a'MA i MB - noxu.ni, OA i OB - ix npoenqii.Hexair IMAO = d, roAi ZMBO = 90' - ctc.
B 'AMO: a6 --*L.sln fl,
""16
-
cM.7.18. Hexa* giarona;ri AC i BD vorrapuryrsaxa ABCD
fieperr4rrarorbc,s n rou4i O.r{epea ABi npaui AC i BD, qo fieper]rHarorbcs, Mo]rd-Ha npoBecrr,r e.4r4Hy nJrorrlzuy o (axc, IIr). Tovxrz .A,B, C i D,'relr(arb y r1ift n.norquui,6o A e AC, C e AC,BeBD,DeBD.
7.19.CeAB,DeAB.{epee 4ri roqrcra C i D sa a}cciolroro I, morxna npo-BecTr4 eAHHy rrpgMy,r{epea nparvri,4S i CD, r1o neperr4Harorbcs B roqqi C,aa arcciouoro II, uoncna npoBecTr4 eArrHy nJroulr{Hy.
7.20. Ilpone4eMo qepeo 4aui npavi a i b n;rorquny o, Ileuorrtna-apo6rarz sa arciol{oo IIr. flpanna c, sra rre-perr,rnae 4aHi npaui, Mae 3 nJron{ilHoro c 4ri cni.nsuiro'ur'u M i N (rovru fieperr,ruy B AaHnMn npaur-ruu).3a uac,ri4xou s axciom (S2,2), sxiqo 4ri rouxr,r npa-moi na.nexarr n.norqrni, ro i scg npsMa c Halrelx[TF,
- n.noulr,rni ct.
7.21. Ilprrvri a i b, axi neperr,rnarorhcs n roqqi O,oAHO3HarrHO Br,rSHaqaroTb fiJrorqrrHy cI.
,{ai rouxz A i B npannoi AB .ne*carr y n,ror4rania,6o A e a, B e b, roMy 3a uac.nigxolr a axcioru(52.2) AB c o. Ananori.rao ArB, c cx,.
Ao = Jill' - Mo" =a'.-, -a" =
s1n- cx
E1l - "t"' ", 4 cos o(=\/ fi'il - ""."
'
3 rBMO: BM = ---:- = o ,
sin(90" - 0) cos o(
Bo = JEM' - Mo" =
AO a cos cl a sin o d cos cr cos oc acos' ct. =___, .BO sinc cos0 dsinq'slnct dsln-cx
,(omnoxrzlto qlrceJrr'Hun i sHaNreHHu( Ha o:
Ao o" "o"b
d et cos'd a" a" AM2
Eo= "'"i"m = .in'o
*T = "itf "
t *"'" = Effi'31. ,{Nn. N, 6.78.
32. 3a r:eopeiuorc npo oproroHaJrbHy npoefiqiro S"o = S. ' cos <p'
s3ni,4cra cosq = +.
,5^
3uafi4enno nrrorqy npoercqii ea Soprry.noro repoua: p = 14 * 1^6 * 6
7.22. Saaacni4norvr a arccioru crepeouerpil uepec rpr4 rorr-xn A, B i C, ari He Jrerfiarb na o4nift npnnnifi, rroxnanpoBecTr,r eAfiHy [Jrorqr,rHy g.To.rru A i B tencarr y n.noulrzui c[, roMy Bcs trpffMaAB c a. Ana;rori.rno AC e a,M e AB, K e AC,rouy M e u i ff e o,, romy.MK c crc,
Xe MK, rouyXe cr.
7.23. Ilpnrvri aib, axi neperr,rHarorbca n'rovqi O, o4uo-SHaqHO Bt4SHaqarOTb nJrotqrrHy 0.Ocxi.nrrra roqrcra ,4 i B ira.nexarr np,fiMrrM a i b, roBorrr Jrell{ar}, y n.norquni o, ToMy npa*ra.A.B Jre*cr,rrr,y n,rorqzni cr..To'rira Y ,'rerrcrrrb na npamifi.AB, rorvry Y e cx,.
DCL
A Bc
M
N
O(
716 i::::: rEOMETPIS Sincxixa O. 9. ra ix. Ara.4emirHufi piBeHb ::#:i:i:::iii:ii [r:{l{iiiii{l$ TEOMETPI9 Eincxixa O. 9. ra in. Ara4euivnrafi pieenr fiii; 217
lly lll)llMtrx ll I I'x llJllrll(rrr!l It llltnMv I t, rlrrlr!rr1'rhrrY/1. Artr',tr.ltr,:r'trr,ttcv ('M(,;l(llrr lllx,ll(( tll Il.Jll.l\ll (r.llltV
npfiMy, napaJlenrHy b. A 3l{atlurb, ltl)tlMtl ('r6itrttlrl,-
cf 3 rrpgMoro cr, a orlrce, re?tcffTb y uouluui u.Taxult uuuolt, 6y4s-arca np,gMa c, nap€rJrerbHa b, axa neperuHae npfiMy a' Jlo)lcn'rl'
y n,rorqrni c.
7.25. [us. aa.qaqy J\! 3.38.
7.32. All(:l) ulll,l./tr'.lot,l,rM, () ,11y,11q11 rr.l,rrr,nlrv iiorrr 2lirrrrOr:rrirn,nn MA - M(:, ,r,o r AMC pillo6c4perrnii.lltr u.nacrueicrro trapalerorpaMa AO = OC, rcgi MOnc4iana, a BHaqrrrr,, i encora IAMC. MO LAC.Anaaoriuno, MO * n:,z,cora pieuo6e4penoro rpr{rcyrnr,rftaBMD (BM = DM, BO = OD\. MO L BD.Ocxi,rsxr.r npxwa MO nepnenAzxyJ'rflpna 4nou npairnum BDi AC, rqo nepeurnarorbcs, rrJrorr{zHn (ABC), ro Ba osnaxororrepneHAuxyJ'rflpnocri npnuoi i n.norqnuu MO L (ABC).
7.33. [iarona,ri xnagpara ABCD neperranarorrcs n rourli O:AC I BD,AM I (ABC), AO L BD, rogi OM L BD sa reopeMoro nporpu nepneHAr,rKyrsp[.OcHirrrcrr npama BD neptreHAr4nyrspHa ABoM rrpflMxM,rqo fleperrrHarorrcs, rrrrorqr{Hr (AMO\, ro BD L (AMO) sa .
oJHaKoro fleptreHAhKyJrspnocri npamoi i naorursn.
7-34.Hexaft piaui n.rrorqzuz o i p nepneHgzlcy.napni go npauoi a. Ocximxu uepeaToqr{y npoxoArrrb eAlHa trJrotquHa, neptreuArrnyJrspna 4anifi npamifi, trJrorqrrH[ di B neperuuaorr fipffMy a n pianrzx roqrax A i B. flprznycruMo, fiIo firoquxlr o i Buarorr, cni,rrHy routcy M. flpoaegerro rrJ'roquHy y qepea rrp.rrMy a i.,rouxy M . Ocxi,rr-n[ trpsMa a nepleHA[KyrspHa nror{uAaM o i F, To y leperr,rnae flJaorqtrHrr d i Fno trpsMilM MAi MB, nepneHnrrcyrapHnm npslriri a. TarcNm qrruoM, y nrorqzni yr{epea roqry M npone4eni 4ei piaui npami, nepneu4lrryaapHi o4uiii npauifi a, rqoHEMOXJ hBO.
Orrxe, nrorqznrt c[ i B He uarorr cni.ntnux roqon, ro6ro napa.ne.irrni.
7.35. ABCD - rerpae4p. M cepe4uuaBC,AB=AC,DB=DC.dABC - piniro6e4peHufi, AM - fioro ue4iana, ro1i AM -Br4cora, AM I BC.r CDB rasotx pinuo6e4pennii sa yMoBoro i DM - fioro me4ia-na, a ehavr.rrr,, i eficora. DM L BC.Taxulr unHonr, nparva BC treptreuAr.rlcy,'rflpHa ABoM rrprMr4MAM i DM rrnouquwu(ADM). Ba ogHarroro nepneu4ur<ynaprocrinpauoi i nrrorqr?rHn BM L (ADM).
7.36. {nn. Ni 4.15.
7.37. AN L(ABC). fl.norquna (/DN) npoxo4nrb qepeo np,n-ny AN. Toai (-4rN) L (ABC).FD L (ABC). If.norquna (FDC) npoxogurb qepea rpfiMyFD. Toai (FDC) t (ABC\.Hexaft BK - Bucora ilapaierorpaMa.BK L AB, EB I BK Ba orHaqeHH,sM hepneHAnKy,r.spHocrirpffMol i trrorrluHu. To,qi BK L (ABE).BK L DC, FD L BK sa oBHarreHHsM nepneHArzrcyJrspuocri npar,roi i naorquun. TogiBK r(FDC).Il;ronlnuu (ABE) i (DFC) nepneH4zxyanpHi npauift BK, roti (ABE) I @FC\.
7.38, [ue. N! 7.37.
7.39. !rs. nnacrunicrr a $4.2.
i;;iittlii{ti TEOMETPIC EincHixa O. L ra ix. Axa4ervrivHufi piBeHb i{d 719
7.27. SABC - rerpae.4p."lo,txn M, N, P i K - cepeAllHu pe6ep AS, BS, BC i ACnignoni4Ho. I{i rourcr - Bepilu Hu flpocropoBoro qorrrptr-
KyrHuKa. V sanaqi 7.26 6ytto Ac:BeAeHo' rqo MNPK - na-pareJrorpaM, MP i NK - fioro giarouali, roMy BoHu rre-
perilHarorbcff i ro.rxoro ueperuHy 4irarr,c,s nasni,a. AHaro-riqHo AoBoAr4Tr,ca, u1o ri4pisxz, rqo e'e4uyrorb cepe,4[Hupe6ep,4-B i SC taAS i BC, neperruarorr'cff B rii rrce rouqi.
7.28. !ns, MoAyrrb 3, Tecr 4ra camonepenipnrz Ne 31.
7.29. Muuo6ixni npaui a i b,rexcarr y piaHux nJrorqrrHax d i F.Hexaiic( d,bcB.Ilpone4emo upflMy c ll b, srca treperr4Hae trpsMy a e rovqi C.
ToAi np,sui a i c o,quooHaqHo BlrsHaqarorb nJ'roquHy.Biasmeuo ua npauifi b 4oni.nury rouHy B i npone4euo-ue-per Hei npauy a, ll a.3a osnaxorc napane,rr,uocri rrJrorqr.rn o ll P.9epes rouny C wroxua npoBecr[ eArzHy nJrorrluHy, irapa-reluy p.
Ilpone4euo inury npsmy d, rqo6 nona 6y.na napale,'rrnanpflMy o. Ilparui c i d ,rerxars n o4niii n,rouqzHi. Ilpama a
roqrtrr, a orr(e, Jlexcrrrb r o4nifi n.norquHi.
7,3O, BepulznuAi B napanenotpauaABCD, a orxce, i npa-ua -4-B ;rexcuri y nrorqvni cr.
BepurrHa C ne .nexurr y uiii.CD I AB Ba oaHaqeHHEM rraparerorpaMa."foli. CD ll ,43 sa oauariorc trapaJreJrr'Eocri npcmo'i i n.no-
lrlrrHt4.
7.31. flplrnycruMo, rqo npaui a i AB ne napa.nemni. To4i moxc.nrani ABa Buna,qftn.
1) Ilpsui aiAB - uuuo6imsi. Togi sa osnaxoro uuuo6i*cnnx trpflMux ($3.1) lpa-Ma a neperrruae nJron-lrrny, s sxifi rexurs npaua.A3, rqo cynepevzrl yuoni.
718 ::i::: rEOMETPI9 EinqxiHa O. 9. ra ix. Axa4enisHrt piBeHb iii:i;rLiii;it
npsuifi b'i neperr.rnanaMae B HrrMn ABi crrirbHi
7.41.
7.42.
7.43.
7.44.
$7.2. flepneHAlrKynep i noxrra Ao nnoulllHll, ei4craxi Ta Kyru
y npocropi7.40. t.
7.57. MO L a, MA, i MB - noxlr.ni, AO i BO * ix npoexqii.MO = 1,2 cu - ei4craur rig rovru M go urorqrzuu cx,.
7.58. ABCD - porvr6, AC a BD = O.AC = 16 crw, .BD = 12 cru.
MO I(ABC). Ba yuonoro rourca M pinnoeiAAaiena niA cropin polr6a, rogi O - qenrpBIII4CAHOTO IIOJTA.
F, N, P,1( - r:oqru Aoruxy rio,aa ,qo cropiu pou6a, ON L AB, OF L BC, OP L CD,OK LAD.To,qi MN L AB, MF L BC, MP L CD, MK L AD sa reopeMoro npo rprr flepreHArr-KyJrspH,MN=MF=MP=MK=8c}.tr.Poar.ngneuo ICOD. OC I OD (4iarouaai pon6anepnen4rzny,rapni).
oc=L,+c=8 cu: oD=LBD=6 cu.22cD = Joc\ oD'=
"61 6' - u/ioo = ro "*.
11s^"^"=:oc-oD=:.8.6=24 (cm2). A22
t =
2So,oo -2 24 =4.g k*\.So.oo = : DC .OP: OP = O" lO3 a MAP MPz = MO2 * OP2;
7.59. ABCD - poiu6, AC a BD -- O, AC = 16 cu, BD = 12 cyr.MO f (ABC).3a yuosoro rouxa M pinaorig4a.neua ni4 cropiH pou6a, ro4i O * ilenrpBnucaHoro Kora.F , N , P, K - ro.rxlr ,4orlrxy rcoJra Ao cropin pom6a,
ON LAB, OF I BC, OP LCD, OK LAD.Toai MN LAB, MF L BC, MP I CD, MK LAD3a TeopeMoro npo Tpr{ treptreHAurcyrspE.MN=MF=MP=MK;MO=2cll.PoarrsHerro aCOD. OC LOD (piaronasti pona6a
nepnen4unyrnpni).
oc=7.ac=8 cu: on=l-nn=6 cu.22cD = ^{oc' * oD" = Js' + 6' = .r[oo = to c*. A
A.
B.
t.
7.45. A.
7.46. D.
7.47.8.
7.48. B.
7.49, Bilnoaila: 3.6 cm.
7.50. B.
7.51. A.
flrcrqo roura 3naxoAr{Tr,cg nooa rrJrorr{[Horo MHoror{yruuxa i piaxori44aaena ni4ycix ftoro cropin,. ro ocuoBa rreprrerAtrr.yJrfrpa, npoBeAeHoro a 4iei rovxz Ha [Jro-rrIuEY, - qeHTp KoJra, BrrncaHoro B MHororcyTHr4n.
7.52. MO L (ABC), rowry MO LAC, MO L BD.ZMAO=tMBO=6O".DB = ABJ, = 8.,0 crvr; OB = 4r[i cm.
Ma= o!,, =j3= a.,E (cru).cos 60o I
Bi0noaidv: [,.
7.53, {un. J\! 6.69.
7,54, AC = 5 cir,r (err.rnercrxuft rpuryrnux).
3 .AMC: twC = AC.-.
= 10 cv.cos ti0"
3 tMBC:MB = ,[MC, - BC' - JIn - a'?= Jioo - 16 =
= 1EI = zvET (cu).
11^DS...-, = 1 MB. BC = i +.zrln= +rDi (cm').22Bi}noeilr': !,.
7.56, MO L P. MA i MB - noxra.ni, OA i OB - ix npoexqii'MA = 73 cv', MB = 37 clt.Ocxi.nrnr.r MA < MB, ryo OA < OB,OA=xcviOB=7xcv'3 yAMO MO2 = \32 - x2 = 169 - x2.
72O ii::: |EOMETPIS Sinsnina O. 9. ra ix. Ara.4ewriqHui piBeHb iiliiiiiiiii i:iaiiii:;i{f::i FEOMETPI9 Fincxixa O. 9. ra iu. AxaAeMi,{xr,ril pieenu tt!; 721
7.60. ABCD - pinno6ivua rpaneqia (AD ll BC, AB = CD).P*"r,= 48 ev, 1A= lD = 6O".MO I o. - ei4cranr ri4 ro.rxu M 4o nnorquuu o,. Toqna M pisuosi4Aarena ri4 ycixcropiH rpaneqii. Ilpoeegeuo MF IAB, MK IBC, MP LCD, ML LAD.- ri4craniui4 rouxu M Ao cropin rpaneuii.OFL AB, OK I BC, OP L CD, OC L AD sa reopeMoro npo rpt4 nepreuAarcyrrspu.OF = OK = OP = OL nrc npoexuii pingnx noxranx.Tarrau vranom, O - qenrp HoJ'Ia, BlucaHoro n rpare4iro, OF = OK = OP = OL -fioro pa4iyclr.
V tBOC IBOC,= I8O" - 2IOBC = 180. - 2 . 80" =3a reopeuoro xocr,ruycis:BCz = BO2 + OCz - 2BO . OC. cos 120..flognaqlruo BO = OC = x cw.lQz = yz t, ,z 2 . x . x. cos(180" 60.);IOO - 2x2 - 2x2 cos 60";
Ioo = 2x2 + z*' .!; Bx, = tao; x =10
F (r > o).
cvr. 3 IAOB AB2 = AO2 -t OB2;
Ao -- 'IAB" - oB' =
Bo=oc=#
EooVa =10 (cnr).
1o!6
I{exrp nora, orrJ4canoro HaBKoJro pinno6ivuoi rpaneqii, ire}fiI{Tb na nepetzui ocicrauerpii rpareqii B cepeAr,rHrruM nepnen,qraKyJrEpoM, npoBeAerruu go 6ivnoi cro-poHr4.
7,61. Tpaneqis ABCD Btrzcana B (oJro. Ocni;ruclr 4yru,gaxnoqesi uix naparelsnuMlr xopAaMrr, pinni, ro pin-si i xopAra, axi ix crnryrorr,.Ocriarxr.r BC ll AD, ro AB = CD. Tpaneais ABCDpieHo6ivna.Hexafi O - qeHTp onacaHoro xora. 3a yMoBoro ocrroBaBC crarye 4yry n 60", ro6ro 4enrpa.nsnwfi xyr BOC
4opisuroc 60". OB = OC stu paaiycn orrucaHoro KoJa. A
Pinno6egperrzfi rprrnyrrir.rx BOC a rcyrou.60" rpu nepurzni e pinnocroponuiu:BO=OC=BC=16cu.Ilponeseuo MO L (ABC), MO = 12 cv.MC - r.i4cranr eig rovxn M go eeprrunr.r rpaneqii. 3 aMOC:
7,62. AO L a, AB i -4C - pinni noxrari, BO i OC - ix npoexqii Ha nrorqr{Hy cx.
Ocxittxu AB = AC, ro BO = OC. 3a yuororc BC = lO cv, ZABC = 6O", lO BC = 30".V pisHo6e,qpesouy a, -BC rcyrrr npr4 ocnoni 4opienrororr 60', rouy aABC - pie-HocropoHnift. AB = AC = BC = 10 ctu.
722 iitl rEoMETPlg Einqxixa o. 9. ra ix. Ara4evrivHr,ri piBeHb i:;l!ii;i;i,!iiii
flpone4enro BH LAD - Bucory rpaneqii. BH ll KL, BH = KL.Ocninsxu 3a yMoBoro r rpaneqirc Mo,fiHa Btrrrcaru KoJo, To
BC + AD = AB + CD =!r*"o = 24 cv. AB = 24: 2 = 12 (cu).
B .ABH: BH = AB.sin 60. ="f= 6Jd (crvr).
oL =: KL =! ", = 6f
= eJe t"*1.222t-
xav =,!ont' u oL' = f,l' t (lG)" ='t0. n = 1136 = 6 1cm;.
Bilnoei}a: 6 cm.
Bionoeiou: to-6 "*.o
7.63. B ro.mz-A Ao nJ'roilIr,rur,r o npoBeAeHo nepleH,4anyJr.flpAO i noxzli AB i AC, OB i OC - npoexqir quxrroxnJrux rignosigno..48 = 13 cm, OB = 5 cw, BC = 19 cu, ZBOC : I2O".3 aAOB AB2 = AOz r OB2:
Ao = ,IEB, - oB, = u{FJ' = v!69 ,t =
= JM4 = 12 (cu).3a reopeuoro xocrznyciri s a BOC:BC2 = BO2 + OC2 - 2BO .OC. cos 120.;192 = 52 t x2.t 2. 5. r. cos 60"; 861 : 2b r f * bx:rc2 + 5x -336 = 0; r, = 16, xz= -21 - rre aa,4oBoJ,rbHfie yMoBy zagati. OC= L6 clr,3 ^Aoc Ac, = Ao2 r oC2; AC = "[r-1d = Jn4 + 2s6= J400 = 20 (cu).BiOnoai1'a: 20 cm.
7.6.4.V yABC AB = 15 cru, AC = lB cu, CB = 14 cu. ^tAM I (ABC), AM - 76 cw. .-'
flponegeruo AK I BC, roni MK I BC aa reopeMoronpo Tprl neprreHAr,rriyJrspE.3naft4erro S"*" B& OopMyroro lepona: A
7.65. y yABC AB - 17 cryr, BC - 8 cu, AC = lb cw.Hannporu rrenuoi cropoHr4 BC lerrrzrr uennlrftxyr A. flpone4emo MA L (A_BC). MA = 2O cw.172 = 289, 152 + 82 = 225 + G4 = 2a9.Maewo: AB2 = AC2 + BC2.I-{e oaaavae, u1o a ABC -
Anpauonytnzri. AC L BC, rogi MC t BC aa reope_MOrO trpO Tpu uepfieHAnr(yr.flpu.
BC 14
(cn).
M
2O cyt
100 - 100
=
12 (cu).
i::ii:t{i*ii FEOMETP|C Einsrixa O. 9. ra ix. Ara4eruivxui piaexr i:ji: Z2g
B .AMC MC2=AM2 +AC2; MC=JrO" -15" =J40o+2%=16t5= 25 (crvr).
Bidnoei}a: 25 cw.
7.66. dABC - np,nMoryarvft (ZC = 9O"), AC = 18 cru, BC = 32 cru'
O - cepeguua rinorenyau, MO L (ABC), MO = L2 cw'
IlposeAelro OK LAC, OL L Bc,roai MK LAC i ML L BC
Ba reopeMolo fipo rprr rrepflexAr{r(yJr.fipu' MK i ML -ni4crani rig ro.tcn M 4o rcarerir rprrriyrrrrrfta.Ocnimnu ABi flpsMi, trepneHAllltyJrspni rperifi, napa- A
.ne.rrni uix co6oro, ro OK ll BC, OL ll AC.To4i aa reo-perroro (Dareca OK i OL - cepeArri trittli rABC.
7 .72. Pinso6etpeni rpranyrnrasu ABD i ACD uarorr cnilrnyocrony -4-D, AD = L6 cm, AB + 17 cm, ZACD = 9O'.flpone4euo qepes rrpsMy BC n;rorquny (BCE\ L AD.Toai BE I AD, CE L AD, ZBEC = 60' - ryr mirxurorqrzuaMrz (ABD) i (ACD).Ocximxu n pinuo6egpenouy rprftlnrrnr{y Bucora, npoBeAe-
Ha Ao ocHoBrt, e rrne4ianoro, ro AE = ED =! AD = 8 crvr.
EC= M =9=8 t".r.ts45" I3 r BCE Ba reopeMoro xocranycin: BC2 = BE2 + CEz - zBE . CE .. cos 60o;
BC= = Jnb + 64 - rn = .i{og = 18 (cu).
Bi0noei6a: 13 cv.
7.73, Pirno6e4peni rpnxyruaxw ABC i /J)C nnarorr cni.nrnyocuonyAC,AC = 16 cm,-48 = 17 cm, BD =13 cw, ZADC =90".flpoxegenro qeper rp,flMy BD n;rol4zuy (BD-O), nepnen4ra-nyrr,rrpry go npruoi AC. Toai BE L AC i DE L AC.IBED = cx, - nyr ui}lt u,rorqunannz rprzxyrnnrcin. AOcxiruxu r piauo6egpenoMy rprrr{yrHuxy Bzcora, npoBe-
E
BD = $z'm -- JT44+289 = n[43 (cNr).
Ocxirrnra AD L (ABC\, AC L BC, ro DC L BC3a TeopeMoro ripo Tpr,r nepilerrAr4rcyJr,fiprl.
7.68, yABC - pionocropounift', AB = BC -- AC = 12 cur'
To.nca M pinnoeig4a.nena ni4 cropin aA-BC. I{e oaua-qae, trIo ocHoBa rrepneunrryJrfipa MO - qerrrp Rora'BrrucaHoro B Tpzrcyrurrrc' Toqxa O nencr,rrr na neperuuiue4ian i nrzcor \ABC. OS = OP = ON - pa,4iycr KoJIa. aMS = MP = MN - ni4crani nig tourclr M go cropinTPITICYTHIIICA' 6O MP L AB, MN L BC' MS I AC SA
TeopeMoro npo Tpr{ flepnerrAr4rtyJrfipi'r'
AB 12 f t;(rs=r= =9!9=2J5 (cu).
2.13 2J3 3 .rreHaAoocgoBn,cMeAiaxorc, ro AE = I oC = * = 8 (cu).22 c
MS2 = MO2 + OS2; MS =
Bi\noei\a: 4Jd cu.B aABE BE =.tEz" - an" =,lt l" - e, =.@g 44 = Jzx = uY aADE IDAE = 45", DE = #= AE = 8 (crvr).
19. 3 roqxr A Ha rrJrorqrrny cx npoBeAeuo neptreHAr4nyJrff p AO i noxnny AB.AO = 12 cv, AB = 13 crr.flpone,4emo OC I AB, rogi AC - npoerqia nepreEArxyJrflpa AO na noxutry AB.B aAoB oB =,tqn, - nO = lTB, -n, - b (cu).Hexafi BC = x cv, roni AC = (18 - r) cu.3 a BCO CO2 = OB2 _ BCz; COz = 52 , x2 __ 25 _ x2.3 IACO CO2 = AO2 - ACz; QQz = 122 - (13 - x)2
23. Y yABC AB = 65 cm, BC = 75 cw, AC = 2O cw.BC ..._ 6i.nrrua cropona, rovy /A - 6i.nrrrr,rft xyr'MA I (ABC), MA = 60 cv.flpoae4ervro -A1{ L BC, M K L BC aa reopeMolo rpoTprr nepneHAr4Ryrtrpr4.AK i,MK - siAcraHi siA xiaqin nepnen4rrxyrrspa AMA 4o 6inl-rtoi croponra BC.3uafi4erro,41{ a tpoprty.nz n.nor4i a/BC:
s .^- = l-qx .BC. Ar( =2s*'2BCIlnoqy rpt'txyrurrr(a snaftgerro Ba OopMy,roro lepona:
E
s*"=@=J8o.1b.b.6o==.ll,r. 5. B - * b., "B r. s = 4. E. b. 2. B = 600 (crra).
,a,x =2 '6oo
= 16 ("*).75
B .AMK MK = "[MA' * AIC = Js600. "56
= ft856 = Jl6'% - +J241 (cn).
Bi|noai\u:16 cu i 4J241 cru.
24. ff.norqa pov6a ABCD 12O cutz, a fioro ctopoua - 12.cw.S =,4-B . KP, ge AB - cropona, a KP - ancora pou6a.
KP=+ =ry =lo (cm).AB 12
To.ma M piruonig4arena ai4 cropiu pom6a, rolry oc-HoBa nepneHAuxytapa MO - rIeHTp Bnr4caHoro KoJra
O. Padyc rco.na r 4opianroe no.noelrHi Br,Icoru: A
27. IltrouTunu cl i.B nepueu4uxy.napni, c - rrp,fiMa rx nepernny. A e s, B e F.3 rcin4ir niApiaxa,4B npoaegemo BC L c, AD L c, rogiBC L a, AD L B. Bilpiaxir AC i BD - npoex4ii ri4piexani4nori4no Ha nrrorr{r4nn a i $, AC = 3rE cm, BD = }JScrvr,CD=3cru.Ityru BAC i ABD - rcyru, yreopeHi nigpierorvr:43 s nno-uquaamlr c i B.
B aBCD BC = ^tED" JD, - Jn4 = v[S = B.,E (cnn).
3 o ACB te tcAB = E9 =3+ = Li ZCAB = 45a." AC 3J2AB' = JAC'z.'+ Bc'z =
"q8.l- ts = ."6a = e 1cu;.
3 a ABD cos.lABD = BD
= 3J5
= -\6 : IABD = to'-AB6Bi0noai0a:45'; 30".
28. flaorqranr,r oc i B nepnen4ury.napni, CD - nparraa fiepert4ny nnorqnn.,43 = 25 cm -Aaruft siApiaon, A e B, B e c.AC L CD ..._ nigcranr nig rouxrz A ,4o fiJrorqr,rHra u, AC = 1"5 cvt.BD L CD - nigcraar eig rouxra B 4o naorqr.rnlr B, BD = 16 civr.3a reoperroro 8 (uo,4yJrr 5) AC I CB, BD L AD.Togi CB i AD - npoexqil ni4pisra -43 oi4noni4no Ha nJro-qunlr o i p.3 *ABC: AB2 = ACz 't BC2;
nc =,[nn' - aCi BC: Jrs" -rb" = JZ00 = 20 (cu).
B oADi: ABz = AD2 + BD2; AD = "[AB" _ BD,,
AD = Jrb' -16' = J625,286 = .,,66e = sJal (cu).
Bi|noeidu:20 cnn, 3i,[T cu.
r{acrrxa 329. Tpareqis ABCD - npsrror yrma, ZA = lB = 9O". AB = 24 cryr, CD = 25 cvt. BD -
6icexrpuca Hyra B. CM L (A.BC), CM = 7JG cv..Ilpoee4euo CK ll AB, toxi CK L AD.ICBD = ZBDA sr< euyrpiurni pianocropouni.To4i IABD - pi'nno6egpeuraft s ocHosop BD.AD=AB=24cw. B3 aCKD (CK = AB = 24 cv): CD'z = CK2 i KD2;
B aAMC: AM =.tmc' o Ae = v?Bb + 86b = vt600 = 40 (cM).Bi1noeidu: 4O etr.
3 aABC BC = ACtgSO" =6 = 2JE (cu)..63
M
65+2O+75p=-=au.
30, Hexafi ABC - 4anzi pirnoctoponnifi rpltryrHr{r{,cropoua rxoro 4opienroe 8 cu..43 ca, C e a.CD L cI, - nigcranr ei4 nepnrraHr.r C 4o nnorqunra cr,
CD = 2JB cnt.Ilpoae4e*ro CK LAB, ro4i DK LAB sa reopeMoro nporpr,r reptreHArrrcy.nspr.r. ToAi ICKD - .niuiftnr.rft xyrABorpaHHoro rcyra ui:*c IABC i rrJrorrlrtuoro c. floana-qlruo fioro g.na apy.rnocri g.
:iiii:;ilt:l! rEoMErPIe Eincxixa O. e. ra ix.
c2JF cu
D)
g
Btx
728 t::# FEOMETPIS SincHixa O. 9. ra in. Ara4er"rivHrfi piBeHb ::.:iiti::.:i::::i Axageuivnun pieenu if{ 729
.t;3 ;ACK (1A = 6O"): CK - AC sin60' = T = a.,E (cu).
srKCD: "inw=ff=2"b 1.4.rE 2' a -- 30'.
31 . ;ABC - pinHo6eApeHnit; AB - AC. Po o",, = 128 cm.
AD - megiaaa (i nucora), AD = 32 cm,
M e (ABC), MA = MB = MC = 65 cw.Ilpoeegemo MO L (ABC) - Bi.4craHb nig rovnu MAo nroulr,(Hrr (ABC). To.qi OA = OB - OC nn npoexqii Apinurax noxurux. O qeHTp Kora, ofir{caaoro HaB-
xoro a,43C, AO = R.IloaHaqr.rNro CD = BD - x crl.
124 2r'loti AC = AB '-"- - --' = 64 r (cM).2
3 "ACD: AC2 = AD2 + CD2; (64 - x)2 - 322 + x'!; 4o96 - 728 x i x2 =l28x=3O72:x=24.CD = BD - 24 cm, AC - 64 - 24 = 40 (cm).
1S.," = - BC . AD -- 24 .32 = 768 (cu").2
3 rfropvvnH s = ob:
rHaiiAevo Ao . R:"4RAB . BC . AC 40 .40 .48
AU = -- . Zb (cMr.4 S.-.u, 4 .768
3 yAOM: AM'1 = AO2 + OM2;
aM =,[AM" -Ao'=,[ss' z'; =J6b-r5X65+r5)=v?0 ,0 =
- Jsooo = 6o (cm).
Bi)noai)a:60 crt.
32, fhoquua, flKa trpoxo,ql4Tb qepe3 npflMy a i rtapa,rensHinpaui,4A, i BB,, nepernua€ nJronluHy tu no npauiri A,Br.A1B1 ll AB, AAt I BBt 3a yMoBorc. Ton' ABBI, trapa-reJrorpaM.AA) -- BBr - 14 cu, Ats = AtBr - 13 cpr, ArB = 15 cu.Srr,"r, - 2S*,r^. SHafigemo S*,u' 3a @opuyaoro lepona.