Геометрия7_9
.pdf§ 14. IIJIOil\i il>iryp |
201 |
128. IIJIOID;I 110,lJ;IBHHX cl>II'YP
Hexali F' i F" - ABi no,ll;i6Hi <l>irypu. 3'SICY6MO, SIK BiAHOCSITbCSI IlJIO~ D;HX <Piryp. OcKiJibKH <l>irypu no,11;if5Hi, TO icHy6 nepeTBOpeH - HSI no,11;i6HocTi, npu SIKOMY <Pirypa F' nepexo,ll;HTb y <Pirypy F " .
Poai6'eMo <l>irypy F' Ha TpHKYTHHKH 6~. 6~, 6 5, ... (MaJI. 303). IIepeTBopeHHSI no,11;i6uocTi, SIKe nepeBO,ll;HTh <Pirypy F' y <l>irypy F", nepeBOAHTb n;i TPHKYTHHKH y TPHKYTHHKH 6~', 6 r , 63 ... poa6uTTSI <Pirypu F". IIJ1om;a <Pirypu F' ,11;opiBHIOG cyMi nJ1om; TpHKYTHHKiB
6 ~, 6 2, 6 ~•... ,a nJiom;a <l>irypu F" ,11;opiBHI06 cyMi m10m; TPHKYTHHKiB 6r, 6 r , 6 3, ....
MaJI. 303
.HKm;o Koe<l>in;iGHT no,11;i6HOCTi ,11;opiBHI06 k, TO poaMipH TpHKYTHHKa 6',( y k paaiB 6iJibllli aa Bi,ll;IlOBi,ll;Hi poaMipH TpHKyT-
HHKa 6 ~. 3oKpeMa, CTOpOHH i BHCOTH TPHKYTHHKa |
/::,. ',( y |
k paaiB |
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6iJibllli aa Bi,11;n0Bi,11;Hi CTOpoHH i |
BHCOTH TPHKYTHHKa /::,. ~. |
3Bi,ll;CH |
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BHDJIHBaG, m;o |
8(6~) = |
k 2 • 8(6~). |
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,IJ;o,11;aBIIlH n;i piBHOCTi DO'IJieHHO, ,ll;icTaHeMO: |
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S(F") = |
k2S(F'). |
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Koecpin;ieHT |
no,11;i6HOCTi k ,11;opiBHIOG Bi,ll;HOIIl0HHIO Bi,ll;IlOBi,11;- |
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HHX JiiHiHHHX |
poaMipiB <l>iryp |
F' i F". Orme, |
nJto14i |
no8i6- |
nux tjJizyp Bi8nocarbca aK K6a8paru ix 6Wn06i8nux Jtiniiinux P03Mipi6.
129. IIJIOID;A KPYTA
HKm;o <l>irypa npocTa, T06To MO)Ke 6yTH poa6uTa ua cKiu11eHHY KiJibKiCTb TpHKYTHHKiB, TO 11 nJiom;a ,ll;OpiBHIOG cyMi IlJIOill; D;HX TPHKYTHHKiB. ,lVISI ,ll;OBiJihHO'i <l>irypu nJiom;a 03Ha1IR6TbCSI TaKHM
'IHHOM.
/.fana tjJizypa Mae nJ£014y S, SIKm;o icHyIOTh npocTi <Pirypu, SIKi MiCTSITb 'i'i, i npocTi <l>irypu, SIKi MiCTSITbCSI B HiH, 3 IlJIO~aMH, m;o SIK
9 KJI8C |
202 |
F
MaJI. 304 MaJI. 305
aasrO,D;HO MaJIO Bi,zi;pi3HSIIOTLCSI Bi,zi; s. 3acTOCy6MO ~e 03Ha'leHHSI,D;JIH 3Haxo,zi;meHHSI nnoi¢ Kpyra.
KpyzoM Ha3HBaGTLCH <l>irypa, SIKa cKJia,n;aeTLCH 3 ycix TO'IOK IlJIOID;HHH, Bi,D;CTaHL Bi,D; HKHX ,zi;o ,n;aHOl TO'IKHHe 6inLwa 3a ,n;aHy. u;.11 TO'IKaHa3HBa6TLCH ijenrpoM Kpyza, a ,n;aHa Bi,n;cTaHL - paoiy- COM Kpyza. MemeIO Kpyra 6 KOJIO 3 THM caMHM ~eHTpOM i pa,zi;iycoM (MaJI. 304).
lllWllftL rcpyui. iJopisH10e no1ioBuHi iJo6yrrcy iJOB:JICUHU rcoJia, 11.fO uozo o6Me:J1Cye, Ha paiJiyc.
,IJ;oBe,n;eMO ~e. Ilo6y,zi;yeMO ,D;Ba npaBHJILHi n-KyTHHKH: |
p, - |
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BnHcaHHH y Kpyr |
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P2 - |
onHcaHHH HaBKOJIO Kpyra |
(MaJI. |
305). |
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MHoroKyTHHKH Pi |
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P 2 e |
npocTHMH <l>irypaMH. MHoroKyTHHK |
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p, MiCTHTLCSI B Kpyai, a MHOrOKYTHHK P2 MiCTHTh Kpyr. |
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Pa,zi;iycH, npoBe,n;eHi y sepwHHH MHoroKyTHHKa P 1, |
poa6HBaIOTL |
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M:oro Ha n TPHKYTHHKiB, KO)KHHH 3 SIKHX ,zi;opiBHI06 TPHKYTHHKY |
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AOD. |
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ToMy |
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0cKiJILKH |
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OC = AC · |
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SAoD = AC · |
AO cos a, |
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TO |
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Sp1 |
= (nAC) AO · |
cos a = P~ cos a, |
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,zi;e p....,.... nepHMeTp MHoroKyTHHKa P., R - |
pa,zi;iyc Kpyra. AHaJiori11- |
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HO ailaxo,n;HMO nJiow;y MHoroKyTHHKa P2: |
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SP2 = |
nSaoF• |
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AC |
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SaoF =AB· AO= -- ·AO, |
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cos a |
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S |
= (nAC)AO = _..!!!!_, |
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p 2 |
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cos a |
2 COB a |
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0Tme, MHOrOKYTHHK |
p,, HKHH MiCTHTLCH B Kpyai, |
Mae nnow;y |
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Sp1 |
= pR cos a, |
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2 |
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9 KJiac |
204 |
3.,D;oeeAiTb, m,o nJ1om,a napaJ1eJ1orpaMa AOpiemoe A06yTKY iioro CTOpOHH Ha BHCOTy, rrpoBeAeHy AO ~ei CTOpOHH.
4.,D;oeeAiTb, m,o rrJ1om,a TpHKYTHHKa AOpiBHI06 rronoeHHi A06yTKY iioro cTopoHH Ha BHCOTy, rrpoeeAeHy AO ~ei CTopoHH.
5.,D;oeeAiTb, m,o rrJ1om,a TpHKYTHHKa AOPiBHIOe rroJIOBHHi A06yTKY ABOX AOBiJlbHHX HOl'OCTOpiH Ha CHHYC KyTa Mi.>K HHMH.
6.,IJ;oeeAiTb, m,o rrJ1om,a Tparre~i AOPiBHI06 A06yTKY rriecyMH OCHOB Ha BHCOTY.
7.HK Bi..zi;HOCHTbCH rrJ1om,i rroAi6HHX <l>iryp?
8.BHBeAiTb <l>opMyJ1y lIJIOm,i Kpyra.
9.3a HKHMH <l>opMyJiaMH o6qHCJllOIOTbCH lIJIO~i KpyroBOl'O ceKTopa i KpyroBoro cerMeHTa?
• |
3A,Jµ.qJ |
1.,D;oBeAiTb, IIJ,O cyMa lIJIOIIJ, KBaApaTiB, rro6yAOBaHHX Ha KaTeTaX rrpHMOKYTHOl'OTpHKYTHHKa, ,u;opiBHI06 lIJIOm,i KBa,n;paTa, rro6y· ,!l;OBaHOl'OHa rirroTeHyai (MaJI. 308).
2.CTopoHH p;Box aeMeJibHHX p;iJIHHOK KBap;paTHOi <PopMH AOPiBHIOIOTb 100 M Ta 150 M. 3Haii,n;iTb CTopoHy piBHOBeJIHKOl IM KBap;paTHOI p;i.JIHHKH.
3.3Haiip;iTb nJ1om,y KBa,n;paTa S aa iioro p;iaroHaJIJIIO a.
4.y CKiJlbKH paaiB lIJIOIIJ,a KBap;paTa, OlIHCaHoro HaBKOJIO KOJia, 6iJlbUia Bi.Ji; lIJIOm,i KBa,n;paTa, BlIHCaHoro B Te caMe KOJIO?
5.HK aMiHHTbCH rrJ1om,a KBap;paTa, HK~o KO.>KHY iioro cTopoHy a6iJlbUIHTH y 3 paaH?
6.Y cKiJlbKH paaiB Tpe6a aMeHUIHTH cTopoHH KBap;paTa, ~06 :Horo rrJ1o~a aMeHIIIHJlacb y 25 paa?
7.qOMy p;opiBHIOIOTb CTOpoHH npHMOKYTHHKa, HK~O BOHH Bi,!l;-
HOCHTbCH, HK 4 : 9, a HOl'OIlJIOIIJ,a 144 M2?
8. 'tJoMyp;opiBHIOIOTb CTOpOHH npHMOKYTHHKa, HK~O :Horo nepHMeTp 74 ,!l;M, a lIJIOIIJ,a 3 M2 ?
9.IlapaJ1eJ1orpaM i npHMOKYTHHK MaIOTb op;HaKOBi CTOPOHH. 3Haiip;iTb l'OCTPHH KYT rrapaJ1eJ1orpaMa, HK~O IlJIO~a :Horo p;opiBHI06 lIOJIOBHHi lIJIOm,i rrpHMOKYTHHKa.
10. KBap;paT i poM6 MaIOTb op;HaKoBi rrepHMeTpH. HKa a <l>iryp Mae 6iJ1bIIIY lIJIOIIJ,y? IIOHCHiTb Bi,!l;lIOBi,D;b.
11. 3HaH,!l;iTb nJ1o~y |
poM6a, |
HK~o iioro |
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BHCOTa 10 CM, a l'OCTpHHKYT 30°. |
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12. 3Haiip;iTb rrJ1om,y |
poM6a, |
HK~o iioro |
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BHCOTa 12 CM, a |
MeHUia |
,n;iaroHaJIL |
13. |
13 CM. |
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,D;oBe,n;iTb, ~o rrno~a poM6a p;opiBHI06 |
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14. |
IlOJIOBHHi p;o6yTKY p;iaroHaJleH. |
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3aaii,n;iTb CTOpOHH poM6a, 3HaJOqH, ~o |
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H.oro p;iaroHaJli Bi,!l;HOCHTbCH, HK 1 : 2, |
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Man. 308 |
a rrnom,a poM6a ,n;opiBHI06 12 CM2• |
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§ 14. fiJIO~i cpiryp |
205 |
15.IIo,ItiJiiTh ,D;aHHii TPHKYTHHK Ha TPH piBHoBeJIHKHX· 'laCTHHH npSMHMH, lll,O npoXO,ll;STb qepea O,ll;HY BepmHHy.
16!" PoaB'smiTb nonepe,D;HIO aa,D;a'ly, BasBmH aaMiCTb TPHKYTHH-
Ka napaJieJiorpaM.
17.1IoMy ,ll;OpiBHI06 IlJIOW,a piBH06e,D;peHoro TPHKYTHHKa, HKllJ,O iioro OCHOBa 120 M, a 6i'IHaCTOpOHa 100 M?
18.3Haii,D;iTh nJiow,y piBHo6e,D;peHoro npHMOKYTHoro TPHKYTHHKa a rinoTeHyao10 a.
19.y TPHKYTHHKY ia CTOpoHaMH 8 CM i 4 CM npoBe,D;eHO BHCOTH ,ll;O
~x CTOpiH. BHCOTa, npoBe,D;eHa ,ll;O CTOPOHH |
8 CM, ,ll;OpiBHIOe |
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3 CM. 1IoMy ,ll;OpiBHI06 BHCOTa, npoBe,ll;eHa AO |
CTOPOHH 4 CM? |
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20. ,lJ;oBeAiTh, w,o |
CTopoHH |
TPHKYTHHKa o6epHeHo nponopWiiHi |
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,ll;O iioro BHCOT, |
T06TO |
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a:b· |
1 . 1 . 1 |
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c = ha . ho . h, . |
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21.3Haii,ll;iTb IlJIOW.Y piBHOCTOPOHHbOro TpHKYTHHKa ia CTOpOHOIO a.
22.3HaHAiTb IlJIOllJ,Y npaBHJlbHOro TPHKYTHHKa, BilHCaHoro B KOJIO pa,Z1;iyca R.
23.3HaH,Zl;iTb n11ow,y npHMOKyTHoro TpHKYTHHKa, HKllJ,0 iioro BHCOTa ,D;iJIHTb rinoTeHyay Ha BiApiaKH 32 CM i 18 CM.
24.1IoMy ,Zl;OpiBHIOIOTb KaTeTH npHMOKYTHoro TpHKYTHHKa, HKllJ,0 iioro rinoTeHyaa AOPiBHI06 73 CM, a nJiow,a 1320 CM 2 ?
25.Y TPHKYTHHKY ABC: AC = a, BC = b. IIpH a:KoMy KyTi C
26. |
IlJIOllJ,a TPHKYTHHKa 6y,D;e ua:H6iJibWOIO? |
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3uaii,ll;iTb IlJIOllJ,y piBHOOe,D;peHoro |
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TPHKYTHHKa, |
y HKOro |
6i'IHi |
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CTOpoHH AOPiBHIOIOTb |
1 M, |
a KYT |
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MiJK |
HHMH CTaHOBHTb |
70°. |
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27. |
3HaHAiTb n11ow,y napa11eJiorpaMa, |
HKllJ,O :Horo |
CTOpOHH |
2 M i |
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3 M, a |
OAHH a KyTiB AOPiBHIOe 70°. |
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28!< 3ua:H,D;iTb IlJIOW.Y TPHKYTHHKa |
aa |
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CTOPOHOIO |
a |
i npH11erJIHMH |
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,ll;O uei KyTaMH a i |
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29. BHBe,D;iTb d>opMy11y repoua JIBH IlJIOllJ,i TpHKYTHHKa: |
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s = ,/P(p - |
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b) (p - |
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,D;e |
a, b, c - |
AOBJKHHH |
CTOpiH |
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TPHKYTHHKa, p - niBnepHMeTp. |
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30. |
3uaii,ll;iTb IlJIOW.Y TPHKYTHHKa aa TpbOMa CTOpoHaMH: 1) |
13, 14, |
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2) 5, |
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17. |
65, |
80; |
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5) |
13, |
12 |
1 |
6) |
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44 |
1,83. |
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3713. |
4713; |
212' |
375' |
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31. OropoHH TPHKYTHHKa a, b, c. 3Haii,D;iTb BHCOTY TPHKYTHHKa, onyw,eHy Ha CTOpOHY c.
32. Bi'IHiCTOpOHH TPHKYTHHKa 30 CM Ta 25 CM. 3HaiiAiTb BHCOTY TPHKYTHHKa, onyw,eHy Ha OCHOBy, lll,O ,ll;OpiBHIOe: 1) 25 CM;
2)11 CM.
33.IIepHMeTp piBH06e,Z1;peHoro TPHKYTHHKa AopiBHI06 64 CM, a iioro 6i'IHaCTOpoHa Ha 11 CM 6iJibma BiA OCHOBH. 3Haii,ll;iTb BHCOTY
TPHKYTHHKa, onyw,eHy Ha 6i'IHYcropoHy.
9 KJiac |
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206 |
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34. |
3H8HAiTL BHCOTY TPHKYTHHKa, y HKOl'OCTOpOHH AOPiBHIOIOTL |
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13, 14, 15 CM. |
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35. |
3Haii,D;iTL BHCOTY TpHKYTHHKa ia |
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CTOPOH8MH: 212, 3 |
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npoBeAeHy AO OCHOBH 212 . |
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36. |
3HaiiAiTL HaHMeHmy BHCOTY TpHKYTHHKa ia CTOPOHaMH: 1) 5, |
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5, 6; 2) 17' 65, 80 i Haii6iJILmy BHCOTY TPHKYTHHKa ia CTopoHa- |
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MH: |
25 |
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3) 6'6'6; |
3713, |
4713. |
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37. |
3HaiiAiTL IIJIOID;Y Tpanewi'.,y HKOl napaJieJILHi CTOPOHH 60 CM |
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20 CM, a |
HenapaJieJILHi - |
13 CM i 37 CM. |
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38. |
y |
piBHo6i•rniH |
Tpanewi'. OCHOBH |
AOPiBHIOIOTL 10 CM |
i 24 CM, |
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6i11Ha CTOpoHa |
25 CM, 3H8HAiTL IIJIOID;y Tpanewi'.. |
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39. |
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piBH06i11HiH TPBnewi'.6iJILma OCHOBa AOPiBHIOEl 44 M, 6i11Ha |
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40. |
CTOpOHa 17 M, a |
AiaroH&Jlb |
39 M. 3HaHAiTb IIJiow;y Tpane~L |
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,lJ;oBeAiTL, w;o KOJIH AiaroHaJii 'IOTHPHKYTHHKanepeTHHaIOTLCH, |
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TO IIJIOID;a 'IOTHPHKYTHHKaAOPiBHIOEl IIOJIOBHHi A06yTKY iioro AiaroHaJieii Ha CHHYC KyTa Mim HHMH.
41tr ,ZJ;oeeAiTL, w;o cepeA ycix napaJieJiorpaMiB a A8HHMH Aiaroaa-
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JIHMH Haii6iJILmy IIJIOID;Y M86 poM6. |
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42. |
BHBeAiTL TaKi <l>opMyJIH AJIH paAiYCiB OIIHCaHoro (R) i BIIHCa- |
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HOl'O(r) KiJI TpHKYTHHKa: |
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abc |
2S |
+ c' |
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R = 48' |
r = a + b |
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Ae a, |
b, |
c - |
CTOPOHH TPHKYTHHKa, a |
s - |
iioro IIJIOID;a. |
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43. |
3HaHAiTL paAiYCH OIIHCaHoro (R) i BIIHCaHoro (r) KOJia AJIH TpH- |
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KYTHHKa ia CTOPOH8MH: 1) |
13, 14, 15; 2) |
15, 13, 4; 3) 35, 29, |
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8; |
4) |
4, |
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7. |
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44.Bi11Ha CTOpoaa piBH06eApeaoro TPHKYTHHKa 6 CM, a BHCOTa, npoBeAeHa AO OCHOBH, 4 CM. 3aaiiAiTL pap;iyc OilHCaHOl'OKOJia.
45.3H8HAiTL P8AiYCH KiJI, OIIHC8HOl'OH8BKOJIO piBH06eApeHOl'OTPHKYTHHK8 3 OCHOBOIO a i 6i'IHOIOCTOPOHOIO b i BIIHC8Horo B HLOl'O.
46.3Haii,D;iTL paAiyc r unucaaoro i paAiyc R onucaHoro KiJI AJIH piBHOOeApeHOl'OTPHKYTHHKa 3 OCHOBOIO 10 CM i 6i11HOIO CTopoHOIO 13 CM.
47.,lJ;oBeAiTL, w;o B npHMOKYTHOMY TpHKYTHHKY paAiyc BilHCaHoro KOJia AOPiBHIOEl IlOJIOBHHi piamru;i Mim cyMOIO KaTeTiB i rinoTeayaoro.
48.KaTeTH npHMOKYTHOl'OTpHKYTHHKa p;opiBHIOIOTL 40 CM i 42 CM.
3Haii,D;iTL paAiycu onucaaoro i anucaaoro KiJI.
49.,lJ;oBeAiTL, w;o IlJIOID;a MHOl'OKYTHHKa,OIIHC8Horo HaBKOJIO KOJia, AOPiBHIOEl IlOJIOBHHi AOOYTKy nepHMeTpa MHOroKyTHHKa Ha pap;iyc KOJia.
50.'tlepeacepeAHHY BHCOTH TpHKYTHHKa npoBeAeHO nepneHAHKYJIHPHO AO Hei'.npHMy. B HKOMY BiAHOmeHHi BoHa AiJIHTL nJiow;y TPHKYTHHKa?
9 KJiac |
208 |
§ 15. EJIEMEHTH CTEPEOMETPII
130. AKCIOMH CTEPEOMETPII
n;e poaAiJI reoMeTpii, B SIKOMY BHB'laIOTLCSI <l>irypH y IIpoCTopi. y CTepeoMeTpii, SIK i B IIJiaHiMeTpii, BJiaCTHBOCTi reoMeTpH'IHHX<l>iryp BCTaHOBJIIOIOTLCSI AOBeAeHHSIM BiAIIOBiAHHX TeopeM. OcHOBHH.MH <l>irypa.MH y npocTopi e mo11.Ku, npRMi i n.no~unu.
CHCTe.Ma aKCiOM CTepeo.MeTpii CKJiaAat:TLCSI 3 aKciO.M IIJiaHiMeTpii i TpLOX IIpoCTOpOBHX aKCiO.M.
AKcioMH
HKa 6 He 6yJ£a nJ£o~uHa, icHyJOmb mo...Ku, iq.o HaJ£e:Jttamb ifiu nJ£o~uHi, i mo...Ku, 11.Ki He HaJ£e:Jttamb 'iu.
~. HK~o iJBi pi3Hi nJ£o~uHu Ma10mb cniJ£bHY mo...Ky, mo 60HU nepemuHa10mbcJ1. no np11.Miu, ~o npoxoiJumb "epea 1f10 mo"Ky.
C3• HK~o iJBi pi3Hi np11.Mi Ma10mb cniJ£bHY mo...Ky, mo "epea HUX MO:JttHa npo6ecmu nJ£o~uHy, i iJo mozo :Jtt miJ£b1'U oiJKy.
HaBeAe.MO SIK IIpHKJIM AOBeAeHHSI ABOX Teope.M ia CTepeo.MeTpii: 3
BHKOpHcTaHHH.M aKcio.M Ci. C2, Ca. |
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Teope.Ma 15.1. qepea mpu mo...Ku, 11.Ki Ke |
J£e:Jttamli Ha oiJHiu |
np11.Miu, MO:JttKa npo6ecmu nJ£o~uHy. |
TO'IKH(.Man. 311). |
,I(oBeAeHHSI. Hexaii A, B, C - TPH AaHi |
IIpoBeAe.Mo .npSIMi AB i AC (aKcio.Ma I). IIpHMi AB i AC piaHi, 60 TO'IKHA, B, C He JiemaTL Ha OAHiit npSIMiiL IIpoBeAe.Mo ttepea npSI-
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M!IJI. 311 |
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