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Киевский национальный лингвистический университет
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9 KJiac

210

llpJ1.Ma napa;ieJi&Ha n;io~uHi, Jf.K~o 60H.a napa;ieJi&Ha J1.KiuHe6yiJ& npJ1.Miu, ~o ;ie:Neumb y 14-iii n;io~uHi (MM. 315).

ABi n.11o~HHH uaaHBaIOTbCSI napaJLeJLbHUMU, SIK~o BOHH He ne-

peTHHaIOTbCSI.

HK~o iJei napa;ieJibHi n;io~uHu nepemuHa10m&cJ1. mpembo10 n;io~uH010, mo npJ1.Mi nepemuHy n;io~uH napa;ieJibHi (MM. 316). lfepe3 mo""Y no3a iJaH010 n;io~uH.010 MO:NCHa npoeecmu n;io-

~uH.y, napa;ieJibHY iJaHiii, i iJo mozo "'mi;ibxu oiJHy.

BiiJpi3xu napa;ie;ibH.Ux npJ1.MUX, JI.Xi MicmJ1.m&cJ1. Mi:NC iJ6oMa napa;ieJibHUMU n;io~UHaMU, pi6Hi (MM. 317).

a

I

I

I

Ma.JI. 315

MM. 316

~\

 

E

D

Ma.JI.

317

Ma.JI. 318

3 a,z:i; a 'Ia

(8). .n;oBe,ll;iTb, ~o KOJIH

'IOTHpHnpSIMi, SIKi npo-

XO,IJ;SITb 'lepeaTO'IKYA, nepeTHHaIOTb n.11o~HHY ex y BepmHHax napMeJiorpaMa, TO BOHH nepeTHHaIOTb 6y,z:i;i.-SIKY llJIO~HHy, na-

pMeJibHY ex, SIKa He npOXO,IJ;HTb qepea TO'IKYA, TaKO>K y Bep-

illHHax napa.11e11orpaMa.

§ 15. EJieMeHTH c-repeoMerpii

 

211

PoaB'sl3aHHS:.Hexd B, C, D,

E -

TO'IKH nepeTHHY

npS:MHX 3 nnom;HHOIO a, a Blt clt Dlt Ei -

Bi'AOOBiAHiTO'IKH

nepeTHHY npHMHX 3 nnom;HHOIO ~. HKa napaneJILHa nnom;HHi a

(Man. 318). IIpoBeAeMo nnom;HHY y1

'lepea nps:Mi AB i AC.

Us nnom;HHa nepeTHHae nnom;HHH a i

~ no napaneJILHHX nps:-

MHX BC i B1C1• IlpoBeAeMo nnom;HHY y2 'lepeanpSDr1i AE i AD. IInom;HHa y2 nepeTHHae nnom;HHH a i ~ no napaneJILHHX nps-

MHX ED i E 1D1• Oci<:iJILKH 'IOTHPHKyrHHK BCDE -

 

napa-

nenorpaM, TO :itoro npoTHJiemHi cTopoHH BC i

ED napaneni.Hi,

BCll ED. DTme, MaeMo E1Ddl ED, EDll BC, BCll BiC1. 3Bi'AcHaa

BJiaCTHBiCTIO napaneJILHHx npHMHX BHIIJIHBae,

m;o E D

II B1C1.

 

1

1

 

A :a;e oaHa'lae, m;o y 'IOTHpHKyrHHKyB 1C1D1E 1 npoTHJiemHi cTopoHH E 1D1 i B1C1 napaneJILHi.

AHanori•mo AOBOAHTL, m;o y :a;i.oro 'IOTHPHKyrHHKanapaneni.Hi CTOpoHH B1E 1 i C1D1. OTme, 'IOTHpHKyrHHKB1C1D1E1 - napanenorpaM, m;o A Tpe6a 6yno AOBecTH.

132. IIEPIIEHAHKYJUIPHICTb IIP.HMHX

I IIJIOIIUIH Y IIPOCT.OPI

IlpHMi y npocTopi Ha3HB8.IOTLCSI nepnenawcyJLJLpHUMU, Hi<:m;o BOHH nepeTHHB.IOThCH niA npSIMHM KYTOM.

HJC~o np.11.Mi a i b nepneKiJuJCya.11.pKi i a 1, b1 - np.11.Mi, .R.1'i nepemuKa10mbc11., napaaeabKi npRMUM a i b, mo 80KU me:NC nepneKiJuJCy n.11.pni.

IlpsMa, s:i<:a nepeTHHae nnom;HHy, HaaHBaeTi.cx nepnenau1Cy.11.apno10 AO :a;ie'innom;HHH, HKm;o BOHa nepneHAHKYJIHpHa AO 6yAL-HKOl npHMOl, m;o nemHTL y :a;iii nnom;HHi it npoXOAHTb qepea TO'IKYlX nepeTHHY (MaJI. 319).

llp11.M.a, ""a nepemuNae nao~uNy, nepneKiJuJCyaapKa iJo nno- ~uKu, R.1'UfO 801'4 nepneKiJuJCyn.11.pna iJo 080X np.11.Mux y nn,o~u­ Ki, 11.1'i npoxoiJamb 1£epea mo1£1'Y ix nepemuNy (Man. 320).

A

MaJI. 319

Man. 320
AK2 •
= AB2

9 KJiac

212

qepu ''°"'"'Y mo'tKY n.n.OUfUKU MO"'Ka npo6ecmu nepneKouKynapKy oo Ke'i npaM.y, i mi.n.bKU oiJKy. Bci npaMi, nepneKiJuKy-

.n.apKi iJo iJaKo'i n.n.OUfUKU, napane.n.bKi (MaJI. 321).

b

A

a

B

MaJI. 321

 

 

MaJI. 322

Ilepnenawcy.TUlpOM, onyw;eHHM 3 ,ZJ;aHOl TO"llKH Ha ,ZJ;aHy IlJIOW:H-

HY, Ha3HBat:TbCH Bi,ZJ;piaoK, w;o

CIIOJiy<Iae

,ZJ;aHy TO'IKY 3 TO'IKOIO

IIJIOID;HHH i Jiea<HTb Ha

IlpHMiH,

nepneH,ZJ;HKYJIHpHiH

,ZJ;O IIJIOID;HHH.

KiHen;b a;Loro Bi,ZJ;piaKa,

HKHH

JiemHTb y

IIJIOID;HHi,

HaaHsaeTbCH

ocno8010 nepnenawcy.TUlpa.

Biacmann10 Bi,ZJ; TO'IKH,ZJ;O IIJIOID;HHH HaaHsaeTbCH ,ZJ;OBa<HHa nepneH,ZJ;HKYJIHpa, onyru;eHoro 3 n;iei: TO'IKHHa IIJIOID;HHY (,ZJ;HB. MaJI. 320).

3a,ZJ;at1a (16). Bi,ZJ;cTaHi Bi,ZJ; TO'IKHA ,ZJ;O BepmHH KBa,ZJ;paTa

,ZJ;opiBHIOIOTb a. 3Hai!:,ZJ;iTb Bi,ZJ;cTaHb

Bi,ZJ; TO'IKHA ,ZJ;O IIJIOID;HHH

KBa,ZJ;paTa, HKID;O CTOpoHa KBa,ZJ;paTa ,ZJ;OpiBHIO€. b.

Po a B 'Ha a H H H • Hexa:H a -

IIJIOID;HHa, B si<.iit JiejKHTb

KBa,ZJ;paT (MaJI. 322). OnycTHMO a TO'IKHA nepneH,ZJ;HKYJIHP Ha IIJIOID;HHY a; TO'IKaK - ocHoBa nepneH,ZJ;HKyJispa. Hexa:H B -

6y,ZJ;b-HKa BepmHHa KBa,ZJ;paTa. IlpoBe,ZJ;eMO IIJIOID;HHY ~ qepea TO'IKHA, B, K. TpHKYTHHK ABK y IIJIOW:HHi ~ npsMOKYTHHH, 60 npsMa AK nepneH,ZJ;HKYJIHpHa ,ZJ;O 6y,ZJ;b-sKo1 npsMoi: y IIJIOID;HHi a, HKa npoxo,z:i;HTL 'lepeaTO'IKYK. 3Bi,l.l;CH BHIIJIHBae, m;o KB2 - OcKiJILKH B - 6y,ZJ;L-HKa BepmHHa

KBa,ZJ;paTa, TO BCi BepmHHH KBa,ZJ;paTa piBHOBi,ZJ;p;aJieHi Bi,ZJ; TO'IKHK. 0Tme, TO'IKaKe n;eHTpoM 1<.oJia, onHcaHoro HaBKOJIO

KBa,ZJ;paTa.

CTopoHa

KBa,ZJ;paTa ,ZJ;opiBHIOe b, TOMY pa,ZJ;iyc KoJia

R = -!J2·

Bip;cTaHL

TO'IKH A

Bi,ZJ; IIJIOID;HHH KBa,ZJ;paTa

AK= ~AB2 - KB2

= ~a2 -

b;.
324).
323).

§ 15. EJieMeHTH crepeoMeTpii

213

IIoxuJL010, npoBe.n;eHoIO a .n;a1101TO'IKH11.0;i;a1101 nJIOlll;HHH, HaammeTbCH 6y.n;b-HKHH Bi.n;piaoK, HKHH cnoJiyqae ,Zl;aHY TO'IKY3 TO'IKOIOIlJIOlll;HHH i He e nepneH.n;HKYJIHPOM .n;o nJIOlll;HHH. KiHen;b Bi.n;piaKa, w;o JiemHTb y IlJIOlll;HHi, Ha3HBaE:TbCH ocnoeo10 noxuJLO'i. Bi.n;piaoK, HKHH cnoJiyqae OCHOBH nepneH,D;HKYJIHpa i IlOXHJIOl, npoBe,D;eHHX 3 O,Zl;Hiei: i Tie! caMo1 TO'IKH,Ha3HBaeTbCH npoeteu,ie10 noxuJLo'i (MaJI. Ha. MaJIIOHKY AB - nepneH,D;HKYJIHp, AC - noxHJia, BC - npoeKn;i.11 noxHJiol.

TeopeMa (npo TPH nepneH,D;HKYJIHpH) 15.3. HK~o npa.Ma, npoBeiJeua ua nao~uui 1£epe3 ocnoBy noxuao"i, nepneuiJuJCyaa.pua iJo ii npoeKifi"i, mo eoua nepneuiJuKyaa.pua i iJo noxuao"i. I HaBnaKH: a.rc~o npsuia ua nao~uui nepneuiJuJCyaa.pua iJo noxuao"i, mo 60ua nepneH.iJurcyaa.pna i iJo npoeKifi"i noxuao"i (MaJI.

A A

B

MaJI. 323

MaJI. 324

,nBi IlJIOlll;HHH,

w;o nepeTHHalOTbCH, Ha3HBalOTbCH nepnenaU1cy-

JLJlp1tUMU, HKlll;O IlJIOlll;HHa, nepneH,D;HKYJIHpHa .n;o npHMOl nepeTHHY

D;HX IlJIOlll;HH, nepeTHHae .n;aHi IlJIOlll;HHH no nepneH,D;HKYJIHpHHX npHMHX (MaJI. 325).

a

a

y

MaJI. 325

MaJI. 326

9 KJiac

214

HJC~o np1uia a nepneniJuJCyna.pna iJo nno~uKu a, a nno~uH.a ~ npoxoiJumb "1£epe3 npa.My a, mo nno~unu a i ~ nepneH.iJu-

lCYlLR.pni (M8..7I. 326).

133. MHOrorPAHHHKH

l{eozpannuM KymoM Ha3HBaeTLCH reoMeTpHqHa cpirypa, yTBopeHa ,Il;BOMa IIiBIIJIO~HHaMH ia CIIiJILHOJO IIpHMOJO, ~o lX o6Memye,-

pe6poM KYTa. 3a Mipy .zi;BorpaHHoro KYTa npuiiMaeTLcH Mipa IIJIOCKoro KYTa, YTBOpeHoro B nepeTHHi ,IJ;BOrpaHHoro KYTa IIJIO~HHoIO, nepIIeH.zi;HKYJIHPHOlO .zi;o pe6pa (M8..7I. 327).

M nozozpannuM JCymoM HaaHBaeTLCH cpirypa, HKa cKJia.zi;aeTLCH a IIOCJii,ll;OBHO CIIOJiyqeHHX IIJIOCKHX KYTiB ia CIIiJILHOJO BepmHHOIO. CTopoHH IIJIOCKHX KYTiB MHororpaHHoro KyTa HaaHBaJOTLCH iioro pe6paM.u.

y CTepeoMeTpii' BHB1{aJOTbC$1 reoMeTpH11Hi Tina, o6MemeHi CKiHqeHHHM ql{cJIOM IIJIOCKHX MHOrOKYTHRKiB,- MnozozpannuJCu. HaiinpocTimHMH 3 HHX e npU3M.U i nipaM.iau. ,Il;aMo 03Ha'IeHH$1 ~HX

MHOrorpaHHHKiB.

IlpH3M&.

Hexaii ex i cx1 - .zi;Bi nap8..7IeJILHi IIJIO~HHH i P - nJIOCKHii MHoroKYTHHK y IIJIO~HHi a. IlpoBe.zi;eMO qepea ,ll;OBiJILHY TO'IKYx MHOroKYTHHKa P npHMY, nepneH,IJ;HKYJIHPHY .zi;o nJio~HHH ex. Boua 6yp;e nepneH,ll;HKYJIHpHa i .zi;o IIJIO~HHH CX1 i nepeTHe i"i y .zi;e.1ud:H TO'I-

~ X1. reoMeTpHllHe TiJio, YTBOpeHe BCiMa Bi,IJ;pi3KaMH XXi. Haa.1rna- €TbC$1 npJLM.010 npU3M.010 (M8..7I. 328). IloBepXHH npHMOl npH3MH CKJia.zi;aeTbC.f.I 3 MHOrOKYTHHKa p y IIJIO~HHi ex, MHOrOKYTHHKa P1

y

IIJIO~HHi <Xi. HKHii .zi;opiBHJOe MHoroKYTHHKY P,- ocnoe npuaMu,

i

npHMOKYTHHKiB, IIJIO~HHH .f.IKHX nepneH,ll;HKYJIHpHi ,Il;O IIJIOI.QHH

OCHOB ex i

CX1 IIPH3MH, a CTOPOHH, ~o CIIOJiyqaIOTb BepIIIHHH MHOrOKYT-

HHKiB P i

Pi. piBHi. BoHH Ha3HBaJOTLCH 6ittHUMU pe6pa.Mu npH3MH.

Bucomo10 IIpH3MH Ha3HBaeTbC.f.I Bi.zi;cTaHb Mim IIJIO~HHaMH iI OCHOB.

I

I .-l

- -•xr

I

MaJI. 327

MaJI. 328

§ 15. EJieMeHTH CTepeoMeTpil

215

IlpJIMa npH3Ma Ha3HBaE:TbC$1 npa8UJl.bH.OK>, JIKID;O iI OCHOBH - npaBHJibHi MHOrOKYTHHKH.

Ilepepiau IIp$1MOl IIPH3MH IlJIOID;HHaMH, napaneJibHHMH OCHOBElM npH3MH, e MHOroKYTHHKaMH, JIKi ~opiBHIOIOTb OCHOBaM IIpH3MH. IlepepiaH npJ1Mo'i npH3MH nJIOID;HHaMH, nepneH~HKYJIJ1pHHMH ~o OCHOB npH3MH, e Ilp$1MOKYTHHKH. 3oKpeMa, llp$1MOKYTHHKaMH e 'AiaroHaJil>Hinepepiau.

3aAa11a (19). y npaBHJibHiA 'IOTHpHKYTHiAIIpH3Mi IIJior:a;a OCHOBH 144 CM2 , a BHCOTa 14 CM. 3Hait,zµTb AiaroHaJib IIpH3MH.

p 0 3 B'$13 a H H $1. ,D;iaroHaJib npH3MH - :a;e BiApiaoK, JIKHA

CIIOJIY11aE: ABi BepmHHH IIpH3MH, r:a;o He HaJiemaTb OAHiA rpaHi,

HanpHKJiaA BiApiaoK AB1 (MaJI. 329). OcKiJil>KH 6i11Hi pe6pa

AA1 i BB1 napaneJI:&Hi,

To 11epea HHX MOmHa nposecTH nno-

ID;HHY (cx). 't!oTHPHKYTHHKABB1A1 y nnor:a;HHi ex -

npJ1Mo-

KYTHHK, 60 6i•rni pe6pa nepneHAHKYJIJlpHi AO OCHOB npH3MH.

3BiAcH BHIIJIHBae, r:a;o Aiaroaan:& npH3MH AB1 = ~AA:+ A1B:.

AA1 = 14 CM, a A1B1 -

AiaroHaJI:& KBaAPaTa a IIJior:a;e10 144 cM2,

TOMY A1B1 = 12./2.

0Tme, Aiarqaam. npH3MH

AB1 =

= ~142 + (12../2)2 = .,/196 + 288 = .J484 = 22 (cM).

IlpJIMa npH3Ma Ha3HBaE:TbC$1 napa.ne.nenineooM, JIKID;O fi OCHOBH - napanenorpaMH (Man. 330). HKr:a;o ocHOBH npH3MH - npHMoKyT-

HHKH, To BOHa

HaausaeT:&CH npJUto1eym11.uM

napa.ne.nenineooM.

y IIpHMOKYTHOro

napaJieJienineAa BCi rpaHi -

IIPJIMOKYTHHKH.

,!J;OB:>KHHH pe6ep IIpHMOKYTHOro napanenenineAa, JIKi CXOAJITbC$1 B OAHitt BepmHHi, Ha3HBaIOTbCH .niH.iUHUMU 8UMipaMU napanenenineAa

c

\

 

I

 

\

 

I

I

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I

\

 

I

A~---\-

D,

) ---

/

................ \

 

/

/

................ \

 

/

c,

 

 

/

 

B1

 

MB.11.

329

MB.11.

330

9 KJiac

 

216

IlpSIMOKYTHHii: napaJieJienine,11;, y SIKOrO BCi

JiiHiHHi BHMipH

piBHi,

Ha3HBa€ThCSI Ky6oM. 0T'fte,y Ky6a BCi pe6pa piBHi, a rpaHi -

KBa,11;-

paTH.

 

 

llipaM~a.

 

 

,l.J;aMo oaHa'leHHSInipaMiµ;H. Hexaii: P -

IIJIOCKHii: MHoroKyTHHK

y IIJIO~HHi a i S - TO'IKa,SIKa He Jie'fl:HThy IIJIO~HHi a. Cnony- 'IHMo6yµ;h-SIKY TO'IKYx MHOrOKYTHHKa p 3 TO'IKOIOs.

reoMeTpH'IHeTiJio, SIKe CKJiaµ;aeThCSI 3 TO'IOKycix Biµ;piaKiB XS, Ha3HBa€ThCSI nipaMiiJ010 (MaJI. 331). IloBepXHSI ~horo TiJia CKJiaµ;a-

€ThCSI 3 MHOrOKYTHHKa p - OCH06U nipaMiiJu -

i 6i'IHHXrpaHeH,

~o SIBJISllOTh co6010 TPHKYTHHKH ai CIIiJihHOIO

BepIIIHHOIO s -

eepwun010 nipaMiiJu. Biµ;piaKH, ~o cnonyqaIOTh BepIIIHHY nipaMiµ;H 3 BepIIIHHaMH OCHOBH, Ha3HBaIOThCSI 6i'lHUMU pe6paMU. Bucom010 nipaMi,ll;H Ha3HBa€ThCSI nepneH,ll;HKYJISlp, ony~eHHii: 3 BepIIIHHH nipaMi,ll;H Ha IIJIO~HHY OCHOBH, a TaKO'fti ,ll;OB'ICHHa~horo nepneH,ll;H- KYJISlpa. IIipaMiµ;a Ha3HBaeThCSI npa6UJlbH010, SIK~o iI ocHoBoIO e npaBHJihHHH MHoroKyTHHK, a OCHOBOIO BHCOTH nipaMi,ll;H € ~eHTp ~horo MHoroKyTHHKa. TpHKYTHY nipaMiµ;y Ha3HBaIOTh mempaeiJpoM (MaJI. 332). A.7Ie 'laCTOTeTpaeµ;poM Ha3HBaIOTh TpHKYTHY nipaMiµ;y, B SIKOl BCi pe6pa piBHi (npaBHJihHHH TeTpaeµ;p).

s s

MaJI. 331

MaJI. 332

IIepepiaaMH nipaMiµ;H IIJIO~HHaMH, napaJieJihHHMH ocHoBi, e MHoroKyTHHKH, no,11;i6Hi ,11;0 OCHOBH nipaMiµ;H. IIepepiaaMH nipaMi,ll;H IIJIO~HHaMH, SIKi IIpOXO,ll;SITh qepea iI BepIIIHHH, € TpHKYTHHKH.

IlJIO~HHa, SIKa napaJieJihHa OCHOBi nipaMi,ll;H, poa6HBae nipaMiµ;y Ha ,ll;Bi 'laCTHHH(MaJI. 333). Ta 3 HHX, SIKa MiCTHTh BepIIIHHY nipaMi,ll;H, € nipaMi,11;010, no,11;i6HOIO µ;aHiH. ,l.J;pyra 'laCTHHaHa3HBa€ThCH 3pi3aH010 nipaMiiJ010. IloBepXHSI apiaaHOl nipaMi,ll;H CKJiaµ;aeTbCH 3

§ 15. EJieMeHTH cTepeoMeTpii

217

OCHOB - IIO,ZJ;i6HHX MHOrOKYTHHKiB ·y nap8JieJibHHX IIJIO~HHax ~·· i

6i'IHOl IIOBepxHi, CKJia,ZJ;eHOl 3 Tpaneajli. 3piama nipaMi,ZJ;a, HKa YTBOpIOeTbC$1 a npaBHJibHol, TaKom HaaHBat:Thca npaBHJibHOJO. Bucomo10 apiamol nipaMi,ZJ;H Ha3HBaf:TbC$1 Bi,ZJ;CTaHb Mim IlJIO~HHaMH OCHOB.

s

M8ll. 333

 

 

 

 

M8ll. 334

3a,ZJ;a'la (29). y 'IOTHPHKYTHiH apiami:H

nipaMi,ZJ;i CTOpoH:fl

O,ZJ;Hiei: OCHOBH ,ZJ;OpiBHIOIOTb 6 CM, 7 CM, 8 CM i g CM, a MeHina

0CTOpoHa ,ZJ;pyroi: OCHOBH ,ZJ;opiBHJOe 5 CM. 3HaH,ZJ;iTb pemTy CTOpiH

D;iel OCHOBH .

 

 

 

 

 

 

 

Po a B ' a a a H HR. OcHOBH

,n;aHol

 

apiaaHol

nipaMi,ZJ;H -

no,ZJ;i6Hi 'IOTHpHKYTHHKH (MaJI. 334). MeHmi cTopoHH n;Hx

'IOTHpHKYTHHKiB e ni,n;noni,ZJ;HHMH cTopoHaMH. ToMy B ,ZJ;aHoMy

BHila,ZJ;KY KoecpirQ.eHT no,ZJ;i6HOCTi

k = i·

0Tme, pemTa CTOpiH ,D;PY-

rol OCHOBH ,ZJ;opiBHIOe:

 

 

 

 

 

 

 

8 . 5

=

')

 

9 . 5

 

1

-

6

6.s. (cM)

'

6

=

7 2 (c:r.i).

 

 

3

 

 

 

y CTepeoMeTpil BBO,ZJ;HTbC$1

IIOH$1TT$1 o6'eMy ,ZJ;Jl$1

reoMeTpH'IHHX

TiJI. Cnot1aTKY poarJI$1,Zl;aJOTbCR npocTi TiJia, TiJia, RKi MomHa poa6HTH Ha CKiH'leHHY IdJibKiCTb TPHKYTHHX nipaMi,ZJ;. 06'eMH :O:HX TiJI BH3Ha'lalOTbC$1 TaKHMH YMOBaM:H: 1) piBHi TiJia M8l0Tb piBHi o6'eMH; 2) $1K~O TiJIO e o6'e,n;HaHH$1M ,ll;BOX npocTHX TiJI, TO o6'eM ncboro TiJia .zi;opinH10e cyMi o6'eMiB lioro qacTHH; 3) o6'e:M Ky6a a JiiHiHHHMH BHMipaMH, ~o ,ZJ;OpiBHIOIOTb O,ZJ;HHH:O:i, ,n;opiBHIOe O,ll;HHH- n:i. 3 D;HX BJiaCTHBOCTeH o6'eMy npOCTHX TiJI M:aeMo:

9 KJiac

218

1. 06'eM npaM01cymHozo

napa"e"enineoa 3 Jl,£HiUHUMU

8UMipaMU a, b, c 061f,UC"10embCJI. 3a "OPMYlf,010 v = abc.

2.06' £M 6yOb-R.'ICO"i npU3MU 00pi8H10£ Oo6ym'H:y nJl,OUfi ii OCH08U Ha Bucomy.

3.06'eM nipaMiou oopiBHroe ooHiii. mpemuHi oo6ym"'y n"oUfi

iiOCH08U Ha 8UComy.

4.06'eM 3pi3aHo"i nipaMiou oopiBHroe pi3Huq.i 06'eMiB noBHo"i nipaMiOU i nooi6HO"i nipaMiOU, R.'H:a BiOmuHaembCJI. Bio He"i.

5.06' £MU nooi6HUX mi" BiOKOCR.mbCR. R.'IC 'H:y6u "ix BionOBiOKUX lf,iKiUKUX 8UMipi8.

134. TIJIA OBEP'I'AHH.H

TiJLo o6epmannsi - n;e reoMeTpH'IHa<Pirypa, yrsopeHa B peayn1.- TaTi o6epTaHHH nJIOCKOl <l>irypH HaBKOJIO oci, HKa Jie:iKHTh y IIJIOID;HHi <PirypH. HaiirrpocTimHMH TinaMH o6epTaHHH e 1.fUJLinop, Konye i 1CYJlJl (M8JI. 335). l.J;HJiiH,D;p yrsopIOeThCH B peayJILTaTi o6epTaHHH IIpHMOK}"rHHKa HaBKOJIO O,D;Hiei: 3 :Horo CTOpiH HK oci, KOHYC - B peayJILTaTi o6epTaHHH rrpHMOKYTHOro TpHKYTHHKa HaBKOJIO O,D;HOro a KaTeTiB, a KYJIH - B peayn1.TaTi o6epTaHHH niBKpyra HaBKOJio ,zi;iaMeTpa HK oci.

MaJI. 335

IIoBepxHH n;HniH,zi;pa CKJia,zi;aeTLCH a piBHHX KpyriB y rrapaJieJILHHX IIJIOID;HHax - OCH.08 l.fUJliHOpa - i 6i'lHOL no8epxni. IlpHMOJiiHiiiHi Bi,zi;pi3KH Ha 6i'IHiHnoBepxHi D;HJiiH,zi;pa napaJieJILHi oci D;H- JiiH,zi;pa i HaaHBaIOThCH mBipnu.Mu D;HJiiH,zi;pa. Bci BOHH napaneJILHi i MaIOTh ,zi;osmHHY, rn;o ,zi;opiBHIOe BHCOTi D;HJiiH,zi;pa. IIepepiaH D;H- JiiH,zi;pa IlJIOID;HHaMH, napaJieJihHHMH OCHOBaM, e KpyraMH, HKi ,D;O- piBHIOIOTh KpyraM ocHoB. IIepepiaH D;HJiiH.n;pa nnoru;HHaMH, rrapaneJILHHMH oci,- npHMOKYTHHKH.

3a.n;a'la(46). BHCOTa D;HJiiH.n;pa 6 CM, pa,zi;iyc OCHOBH 5 CM. 0 3Ha:H,ii;iTL nnoru;y rrepepiay, npoBe.n;eHoro rrapaJieJILHO oci n;n-

niH.n;pa Ha Bi,D;CTaHi 4 CM Bi.n; He!.

337).

§ 15. EneMeBTH CTepeoMeTpll

 

 

 

 

219

p 0 3 B, $13 a H H $1.

Ilepepia D;HJiiHApa

 

llp.HMOKYTHHK, y

.HKOro 6i•IHi CTOpOHH

-

TBipHi D;HJiiHApa,

a

OCHOBa -

XOpAa

KOJia AB, HKa 3HaxOAHTbCH BiA n;eHTpa 0

Ha BiAcTaHi

4 CM

(MaJI. 336). OcKiJibKH pa.Ai.ye KOJia AOpiBwoe 5 CM, a XOPAB 3Ha-

XOAHTbCH Ha BiACTaHi 4

CM BiA n;eHTpa,

TO AOB)KHHa XOPAH

AB = 2~52 - 42 = 6 (cM).

OTme, IIJiom;a nepepiay S = 6 · 6 = 36 (cM2).

IlpH3Ma H83HBaeTbCH enucaH010 B n;HJiiHAp, HKID;O li. OCHOBH BllHCaHO B KOJia OCHOB D;HJiiHAPB. IlpH3M8 H83HBaeTbC$1 onucaH010 HaB-

KOJIO D;HJiiHAPB, HKID;O li OCHOBH OllHCaHO HaBKOJIO OCHOB D;HJiiHApa. Pa<JiycoM u,uJLiH<Jpa HaaHBaeTbc.a paAiyc Kpyra B ocHoBi D;HJiiHApa.

06'eMD;HJiiHApa 3HaxOAHTb, nopiBHlOlO'lHlioro 3 o6'eMaMHBllHcaHoi" i onHcaHoi" npHaM. IlpH D;bOMY AicTaIOTb cpopMyJiy AJIH o6'lHCJieHHH o6'eMyD;HJiiHApa

Ae rr.Ff' - llJiom;a OCHOBH D;HJiiHAPB, a H - BHCOTa. Bi'lHanoBepXHH D;HJiiHAPB o6'lHCJIJOeTbCHaa <I>opMyJioJO

S = 27tRH,

Ae 2rr.R - AOBmHHa KOJia OCHOBH D;HJiiffApa, a H - BHCOTa. IloBepXHH KOHyca CKJiaAaeTbCH 3 Kpyra B OCHOBi KOHyca i 6i'lHOl

llOBepxHi. Ilp.aMoJiiHiHHi BiApi3KH Ha 6i'lHiHllOBepxHi KOHyca, m;o CllOJiyqaJOTb BepmHHY KOHYC8 3 TO'lKaMHKOJia OCHOBH, Ha3HBaJOTb- C$1 meipHUMU ICOHyca. IlepepiaoM KOHyca llJIOID;HHOlO, napaJieJlbHOlO OCHOBi, e Kpyr. Ilepepi30M KOHyca llJIOID;HHOlO, m;o npoXOAHTb 'lepe3 BepmHHy, e TpHKYTHHK. Bucomo10 KOHyca Ha3HBaeTbc.a nepneHAHKYJI.Hp, onym;eHHH 3 BepmHHH KOHyca Ha OCHOBY.

IlJiom;HHa, napaJieJibHa ocHoBi KOHyca, poa6HBae iioro Ha ABi 'laCTHHH. 0AHa 3 HHX, HKa MiCTHTb BepmHHy, e KOHyc, llOAi6HHH ABHOMy. ,IJ;pyra 'laCTHHaHa3HBaeTbCH apiaaHUM ICOHYCOM (MaJI.

s

Ma.JI. 336

MaJI. 337
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