Добавил:
Upload Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз: Предмет: Файл:

Геометрия7_9

.pdf
Скачиваний:
5
Добавлен:
08.02.2016
Размер:
4.34 Mб
Скачать

O.B.norop€Aon

IlAAHIMETPI51

nIAPyqHHK

AAfl 7-9 KAACIB 3AfAAhHOOCBITHIX HAB11AAhHHX 3AKAAAIB

3amf3epo)KeH.O MiuicmepcmfioM oc6imu i uay-KU YK[Ja'iuu

KMIB

"WKon1p•

2004

N~ 4-492.

BBK 22.151 SI72

II43

3ameepa:HCeno MintcmepcmeoM oceimu i nay1eu Y1epa'inu

(JlHCT MiHicTepcTBa ocBiTH i HayKH YKpai:HH N.! 1/ 11 - 3801BiA17 .09.2001)

IlpoAaata llpHMipBHKiB UhOl'O BHAaHHll

Ilea l'OJJOl'paMH Ba o6KJJ~HHUi E: HeaaKOHHOIO.

HaB'laJibHe BHARHHH

nororeJIOB OJ1e1eciu BacUJLbOeuii

rEOMETPUI

IlnaaiMeTpiH IliAPY'IHHK AJIH 7-9 1rnaciB

aaraJibHOOCBiTHiX HRB'IRJibHHX 3RKJIRAiB

3am8epa:JK:eno MinicmepcmBOM ocBimu i nay1w Y1epa'inu

BiAIIOBiAaJibHHH aa BHIIYCR IO. 0. Kop6yrn .

XyAo:llrne o<t>opMJieHHH A. A. 3iH'leHRa.

IIi,11nHCaHO ,!10 APYKY 10.09.04. <l>opMaT 60x90/16. Ilanip ocjlc. rapHiTypa lllKiJibHa . APYK ocpc. YM. ,11pyK. apK . 15. YM. cpap60Bi,116 . 15,5 . 06JI .-BH.zl . apK. 15,48.

Tupa>K 10 000 np. 3aM.

YBU •IIlKOJillP•.

02094, KHi:B, Byn. Ceprit:HKa, 18. CBi,llOUTBO AK N.! 360 Bi,ll 14.03.2001 p.

BHAPYKOBaHO Ha BAT •Binon:epKiBCbKa KHH>KKOBa cpa6pHKa., 09117, M. Bina I.{epKBa, ByJI . Jlec11 Kyp6aca, 4.

npasa aBTOpie Ta B11AaBH114rea 3ax11~eHi 3aKOHOM YKpa'iH11«npo asropcbKe npaso

i cyMi>KHi npasa» BiA 23.12.1993 p.

APYKOBaHe Konit0saHHl'lKH11r11 a6o 'Ii4acT11H11, 6YAb-RKi iHwi KOHTpacpaKTHi B11AaHHl'l Tl'lrHyrb3a co6ot0 ei,Qnoei,c.anbHiCTb 3riAHO 3i CT. 44. n. 1.3 4boro 3aKOHy.

Iloropt:JIOB 0. B.

II43 reoMeTpix: IIJiaHiMeTpix: IliAPY'I. AJIH 7-9 KJI. 3araJibHOOCBiT. HRB'I. 3RKJI.- 7-Me BHA.- K.: lIIROJIHp, 2004. - 240 C.

ISBN 966-7117-65-0

BBK 22.1511172

© 0 . B. floropenos. 1994 © 0. B. floropenos, 2003.

3i 3MiH8MH

©

BHA8BHHL{TBO •lllKOJillp• ,

ISBN 966711 7-65-0

xy11o>KKe O<jlopMJieHHR, 2004

06'ep,-
I'eoMerpia -

7 KJlaC

§ 1. OCHOBHI BJIACTHBOCTI

HAHIIPOCTIWHX rEOMETPH1UIHX CZ-IrYP

1. rEOMETPlfllHI fl»lrYPH

~e uayKa npo BJiacTHBOCTi reoMeT}>H'IHHXcpiryp. CnoBo .reoMeTpi1u - rpe~Ke, y nepeKJI&Ai ua yRpaiHci.Ry MOBY

oaua11ae •aeMneMipcTBO•. T&Ka uaasa ayMoBJieua aacTOCYB&HHHM reoMeTpii AO BHMipIOBa.Hb Ha Mic~eBOCTi.

IlpHKJI8P,H reoMeTpH'IHHX cpiryp: TJ>HK}'THHK, KBap,paT, KOJIO (M8Jl. 1).

reoMeTpH'IHicpirypH 6yeaIOTb P,OCHTb pi3HOM8HiTHi. q8CTHH8 6yp,b-HKOi reoMeTpH'IHOicpirypH e reoMeTpH'IHOIOcpirypoIO. H8HHH KIDKOX reoMeTpH'IHHXcpiryp e auosy reo11eTpH'IHOIOcpirypoIO. Ha MaJIIOHKY 2 cpirypa a.ni.Ba cKJiap,aeTLCH a TPHKYTHHKa i

Tpi.ox KBap,paTie, a cpirypa cnpaBa - a Kona i 'l&CTHIDIKOJia. Komuy reo11eTpH11Hy 4>iJ"¥PY MO>KH8 YRBHTH CKJiap,eHOIO 3 TO'IOK.

DO

Man. 1

Man. 2

TO'IKHA
d, ....

7 KJiac

EeKJii'A- .n;aem.o-

rpe~bKHH y'leHHil

(JU CT. 'AO H. e.)

4

reoMeTpiR illHpoKO 3aCTOCOBYGTLCR Ha npaKTHaj. Ii: Tpe6a 3HaTH i po6iTHHKy, i iHmeHepy, i apxiTeKTopy, i XYAOlKHHKy. 0AHHM CJIOBOM, reoMeTpiio c~ 3HaTH BCiM. reoMM'piR, ~O BHB'laGTbCR y illKOJii, HaaHBaGTLCR eBKJ1iAoB010, aa iM'RMAaBHborpeD,bKoro B'leHoroEBKJiiAa, RKHH CTBOPHB noci6HHK 3 MaTeMaTHKH niA Ha3BOJO •Ha11aJia•. TpHBaJIHii 11ac reoMeTpiIO BH-

B'laJIHaa ~GIO KHHlKKOIO.

Mu no'IHHaGMo BHB'leHHR reoMeTpii a nJiaHiMe-rpii. llJr.aHiMerpia - ~e poaAiJI reoMeTpii, B RKOMY BHB'laIOTLCR<PirypH Ha

IlJIO~Hi.

2. TO'IKA I IIP.HMA

0cHOBHHMH reoMeTpH'IHHMH<l>irypaMH Ha IlJIO~HHi e TO'l.H:a i npsut.a. To'IKH npHiiHRTO noaHa'laTH BeJIHKHMH JiaTHHCbKHMH 6yKBaMH A, B, C, D, .... IIpaMi noaHa•1aIOTLCR MaJIHMH JiaTHHCbKHMH 6YKBaMH a, b, c'

Ha MaJIJOHKY 3 BH 6a'IHTeTO'IKYA i npaMy a.

IIpaMa HecKiH'leHHa.Ha MaJIIOHKY ao6pamyeMo JIHme 11acTHHY npaMoi, aJie yRBJIRGMo ii Heo6MemeHo npOAOBlKeHOIO B o6HABa 60KH.

IIOAHBiTbCR Ha MaJIIOHOK 4. BH 6a11HTe npaMi a, b i TO'IKHA,

B, C. To'IKHA i C Jr.e:JKarb Ha npaMiii a. MomHa cKaaaTH TaKOlK,

~o i C HaJr.e31Carb npaMiii a, a6o npRMa a npoxoiJurb

11epea TO'IKHA i C.

a

Man . 3

Man . 4

po3ai;uie TO'IKHA

§ 1. Ocuoeui BJI&CTHBOCTi ua8npOCTimux reoMeTpH'IHHX cl>iryp

5

To'IKaB nemHTb Ha rrpsMtii: b. BoHa He nemHTb Ha rrpHMiii a.

To'IKaC nemHTb i Ha rrpHMtii: a, i Ha rrpHMiH b. IlpHMi a i b neperu- H.alOTbC.R. B TO'I~ C. To11Ka C 6 ro11.1eo10 neperuny rrpHMHX a i b.

Ha MaJIIOHKY 5 BH 6a'IHTe,HK aa p;orr0Moro10 niHtii:KH MomHa rro6yp;yuaTH rrpsMy, ~o rrpoxop;HTb qepea p;ui TO'IKHA i B.

Man. 5

0 c H 0 B H H M H B JI a c T H B 0 c T SI M H HaJie)KHOCTi TO'IOK i IIpSIMHX Ha IIJIO~HHi HaaBeMO TaKi BJiaCTHBOCTi:

I. HKa 6 He 6yll4 nplUl&a, icHyHJT'b TO"Ku, UfO Ha11.e:11eaT'b ifirl np1u•iu, i ro"JCJi, UfO He Ha11.e•aT'b iu.

qepe3 6yiJ'b-JUri iJ6i TO"ICU M.031&"4 npot1ecTU npaMy i TiJl.'blCU OiJHy.

IlpHMY MOmea rroaeatiaTH ,IJ;BOMa TO'IKaMH,~o nemaTb Ha HiH. HarrpHKna,11;, rrpsMy a Ha MaJIIORKY 4 MOmHa rroaHa'IHTHAC, a npsMY b-BC.

3 a ,II; a 'Ia (3) 1liu MO)KYTb p;Bi rrpsMi rrepeTHHaTHCJI y ,11;uox TO'IKax?Bi,IJ;rrouip;L rroscHiTL.

Po au'.a a a H H s. HK6H p;Bi IIpHMi MaJIH p;ui TO'IKHrrepe· THHy, TO qepea ~ TO'IKHrrpoxop;HJIH 6 JJ;Bi rrpsMi. A ~e HeMOlKJIHBO, OCKiJibKH 11epea p;Bi TO'IKHMOmHa rrpoBeCTH TiJibKH OAHY IIp.RMy. 0Tme, p;Bi rrpHMi He MO»tYTb MaTH ,IJ;BOX TO'IOKrrepe-

- THHy.

3. BIWI30K

Ilop;HBiTbCH Ha MaJIIOHOK 6. BH 6a'IHTerrpsMy a i TPH TO'IKH A, B, C Ha ~ rrpsMiii. To'UCaB nemHTb Mi.JK TO'IKaMHA i C, BOHa

i C. MomHa TaKom cKaaaTH, ~o TO'IKHA i C ne-

I t{HCJIO B .ll;Y)KK8X 03H8'18€ HOMep 3~aqi B CIIHCKY 3~8'1, naseii;eeux y KiHI~i naparpact>a.

7 KJiac

6

MaJI:. 6 Man. 7

maTi. no pi3Hi 6oH:u Bi.A TO'IKHB. To'IKHBi C JiemaTi. no oiJun 6iH:

BiA TO'IKHA, BOBH He pooAiJIBIOTi.ca TO'IKOIOA. To'IKHA i B JiemaTi. no OAHH 6iK Bi.A TO'IKHc.

BiiJpi3KOM Ha3HBa6T'bCB 11acTHHa npaMo'i, mca cKJiaAaeTi.cs a

ycix TO'IOK~e'i npBMOl, ~o JI8)1(8Tb Mim ABOMB AaHHMH i"i TO'IKaMH. IJ;i TO'IKHHa3HBaIOTbCB KiHi.µiMU eiiJpi31Ca. BiApiaoK noaHa'laIOTb, aanHcyIO'IHiioro KiH~. KoJIH roBOpaTi. a6o nHmYTb oiApiaoK AB•,

TO MaIOTL Ha yBaai BiApiaoK 3 KiB::u;sMH B TO'IKaXA i B.

Ha MaJIIOHKY 7 6a'IHMOBiApiaoK AB. BiH e qacTHHOIO npaMo'i

AB. Il;IO 11acTHBY npaMo'iBHAi.neHo mHpHOIO JiiHieio. To'IK8.X npaMo'iJiemHTb Mim TO'IKaMHA i B, TOMY BOHa HanemHTi. BiApi3Ky

AB. To'IKaY He JiemHTL Mim TO'IKaMHA i B, TOMY BOHa He HaJiemHTL

Bi.Api3Ky AB.

0 c H 0 B H 0 IO B JI a c T H B i c T IO p03M~eHH$1 TO'IOKHa npa-

Miii Ha3BeMO TBKY BJiaCTHBiCTb:

I I. 3 rp11ox ro'Wx Ha npiudu oiJHa i riJJ11rcu o8Ha Jie:11&Ur11 Mi:w

OBOlllQ. iHWUMU.

4. BHMIPIOBAHHH BIAJ>I3KIB

,lJ;JIB BHMipIOB8HH$1 BiApi3KiB KOpHCTYIOT'bC$1 piaHHMH BHMipIO- B&JibHHMH iHcTpYMeHTaMH. HaiinpocTimHM T&KHM i.HcTpyMeHTOM

e JiiHiAKa a noAiJIKaMH Ha Hi.A. Ha M&JIIOHKY 8 BiApiaoK AB AOPiB-

RJOe 10 CM, BiApiaoK AC AOpiBHIOe 6 CM, a eiApiaoK BC AOPiBHIOe

4 CM.

,ll;OB)KHHa BiApiaKa AB AOPiBHIOe cyMi AOB)KHH eiApiaKiB

AC i

BC.

0 c H 0 B H H M

H B JI a c T H B 0 c T $1 M H BHMipIOBaHH$1 BiApia-

KiB H83BeMO TaKi

BJI&CTHBOCTi:

§ 1. OcHoBHi BJiftCTHBOCTi HaiirrpocTimHx reoMeTpH'lHHXcl>iryp

7

A

[

B

MaJI. 8

III. Ko:11eHuu BiiJpiaoK Mae neBHY iJ08:11tuHy, 6i".,,"'Y Bio "Y""· ,lfo831&UHa BiiJpia""' iJopitJHH>e cyxi 0083/eUH 'UJCTUH, Ha siKi BiH poa-

6UBaer'bCsi 6yiJ'b-1'1C010 UOZO T01'1C010.

~e 00Ha11ae, w;o KOJIH Ha eiJJ;piaKy AB eaaTH 6yJJ;L-aKy TO'IKYC,

TO AOB)KHHa BiApi3Ka AB AOPieHI06 cyMi

AOB)KHH ei)l;pi3Kie AC

i BC. ,IJ;oe)KHHY eiJJ;piaKa AB HaaHeaIOTL

TaKo)K Biocrann10 Mi)K

TO'IKaMHA i B.

3 a A a 11 a (9). TpH TO'IKHA, B, C Jie)KaTL Ha OAHiii npaMiii. Bi,l\oMo, w;o AB = 4,3 CM, AC = 7,5 CM, BC = 3,2 cM. lJH MO)Ke To11Ka A Jie)K&TH Mi)K TO'IKaMHB i C? lJH MO)Ke To11Ka C Jie)KaTH Mi)K TO'IKaMHAi B? .HKa a TpLox TO'IOKA, B, C

Jie)KHTL Mi)K ABOM& iHmHMH?

p 0 3 B. $1 3 a H H a. .HKw;o TO'IKaA Jie)KHTL Mi)K TO'IKaMH

B i C, TO aa enacTHBicTIO BHMip10eaHHa ei,l\pi3Kie Mae 6yTH AB + AC = BC. Ane 4,3 + 7,5 =I= 3,2. 0T)Ke, TO'IK&A He ne)KHTL Mi>K TO'IK&MHB i C•

.HKw;o TO'IKaC Jie)KHTL Mi)K TO'IK&MHA i B, TO Mae 6yTH AC + BC = AB. Ane 7,5 + 3,2 =I= 4,3. OT)Ke, TO'IKa C He Jie)KHTL Mi)K TO'IK&MHA i B.

3 TpLox TO'IOKHa npaMiii A, B, C OAHa ne)KHTL Mi)K ABOMa iHWHMH. OT)Ke, ~e10 TO'IKOIOe TO'IKaB.

5. IIIBIIJIOID;BHB

lloAHBiTLca Ha MaJIIOHOK 9. llpaMa a poa6HBae nnow;HHY Ha ABi nieIIJIOUJ;HHH. ~e po36HTT$1 Mae TaKy BJiaCTHBiCTL. .HKw;o KiH~ $1KOrO-He6yAL Bi,l\pi3Ka HaJie)KaTL OJJ;Hiii niBIIJIOUJ;HHi, TO ei)l;pi30K He nepeTHHae npa:My• .HKw;o KiHu;i BiApiaKa HaJie)KaTL piaHHM nieIIJIOUJ;HHaM, TO ei,l\piaOK nepeTHHae npaMy.

Ha MaJIIOHKY 9 TO'IKHA i B ne)KaTL e OAHiii a niennow;HH, Ha aKi npa:Ma a poa6ueae IIJIOUJ;HHy. ToMy BiApiaoK AB He nepeTHHae npaMY a. To'IKHC i D Jie)KaTL y piaHHX niennow;HHax. ToMy eiApiaoK

CD nepeTHHae npaMy a.

10).

7 KJiac

8

Man. 9

0 c H 0 B H 0 IO B JI a c T H B i c T IO poaMim,eHHH TO'IOKBiAHOCHO npHMOi Ha IlJIOlll;HHi H83BeMo TaKy BJiaCTHBiCTL:

W. HpRMa poa6usae 1U011fUHY Ha 8si niBn.1io11fliHu.

3 a A a 'Ia (17). ,D;8Ho npHMY i TPH TO'IKHA, B, C, w;o He JiemaTi. Ha Hiii. BiAoMo, w;o siApiaoK AB nepeTHHae npsMy, a BiApiaoK AC He nepeTHHae ii. 't!HnepeTHHae npsMy BiApiaoK

BC? BiAilOBiAL IlOHCHiTL.

p 0 3 s'H 3 a H H H. Ilpa:Ma poa6HB86 IlJIOw;HHY Ha ABi niBIlJIOw;HHH (MaJI. To'IKaA HaJiemHTL oAHiii a HHX. BiApiaoK AC He neperHHae npa:My. 0Tme, TO'IKaC JiemHTL y Tiii caMiii niBnJIOw;HHi, w;o ii TO'IKaA.

BiApiaoK AB nepeTHHae npsMy. 3Ha'IHTL,TO'IKaB JiemHTL B iHmiii nisnJiow;HHi. TaKHM 'IHHOM,TO'IKHB i C JiemaTL y piaHHx uisuJIOw;HHax. A ~e OOHa11ae, w;o siApiaoK BC nepeTHHae A8HY DPHMY.

6. IlIBilP.HMA

3aAa'la(20) . .IJ:aHo upsMy a i TO'lKHA, X, Y, Z Ha Hitt (MaJI. 11).BiAOMO, w;o TO'IKHx i y JiemaTL uo OAHH 6iK BiA TO'lKHA, TO'lKHx i z Tem JiemaTL DO OAHH 6iK BiA TO'lKHA.

MaJI. 10

MaJI. 11

n03H8'1HTHAB.

§ 1. 0CHOBHi BJiaCTHBOCTi Haii:npocTiIUHX reoMeTpH'IHHXcl>iryp

9

HK poaMiw;eHi TO'IKHY i Z BiAHOCHO TO'IKHA : no OAHH 6iK,

'IHno pi3Hi 6oKH? IIOSicHiTh BiAnoBiAh.

 

Po 3 B'H 3 a H H H. IIpoBeAeMo qepea TO''IKYA

AOBiJihHY

npHMY b, Bi,l.\MiHHY BiA a. BoHa po3i6'enJIOIIJ;HHY Ha ABi niBnJIOIIJ;HHH. 0AHiii 3 HHX HaJie:>KHTh TO'IKax. y Tiii caMiii niBnJIOIIJ;HHi JiemaTh TO'IKHY i Z, 60 Bi,l.\pi3KH XY i XZ He nepeTHHaIOTh npHMy b. OcKiJihKH TO'IKHx i z Jie:>KaTh B OAHiH niBnJIOIIJ;HHi, TO BiApiaoK YZ He nepeTHHae npHMY b, a oT:me, He MiCTHTh TO'IKYA. To6To TO'IKHYi Z Jie:maTh no OAHH 6iK BiA TO'IKHA.

llienpa.M.010 a6o npo.M.eHe.M. Ha3HBaIOTh qacTHHY npHMOl, HKa CKJia).\a6ThCH 3 ycix TO'IOKajsi: npHMOl, w;o JiemaTb no OAHH 6iK BiA ).\aHOl Ha Hiii TO'IKH.u;a TO'IKaHa3HB86ThC.R llO'UlT1'06010 TO'l/COIO niBnpaMoi:. Pi3Hi niBnpaMi OAHis1 ii Tisi: caMoi'npaMoi'i3 cniJihHOIO no'laTKOBOIOTO'IKOIOH83HBaIOThC.R iJon06H.R.JtbHU.M.U.

IIiBnpaMi TaK caMo, .RK i npaMi, no3Ha'laIOThC.RMaJIHMH JiaTHHChKHMH 6yKBaMH. Mo:mHa no3Ha'laTHniBnpJ1My ABOMa TO'IKaMH: notJ:aTKOBOIO i w;e HKOIO-He6y,l.\h TO'IKOIO,HKa HaJie:>KHTh niBnpaMiii. 1Iot1aTKOBY TO'IKY cTaBJI.RTh Ha nepmoMy Micaj. HanpHKJiaA, niBnpaMy, HaBe).\eHy :>KHPHOIO JiiHi6IO Ha M8JIIOHKY 12, MO:>KHa

a

A

8

 

 

MaJI. 12

3 a A a q a

(22). Ha Bi,l.\pi3Ky AB B3.RTO TO'IKYC. CepeA

niBnpHMHX AB, AC, CA i CB Ha3BiTh napy niBnpHMHX, w;o 36iraIOThC.R, i napy AOnOBHHJihHHX niBnpaMHX. Bi,l.\noBiAh nO.RCHiTh.

P o 3 B'H 3 a H Ha (MaJI. 13). ,lJ;aHi niBnpaMi MaIOTh no- 'IRTKOBOIOTO'IKOIOa6o TO'IKYA, a6o TO'IKYC.

Po3rJiaHeMo cnoqaTKY niBnpaMi a no'laTKOBOIO TO'IKOIO

A (niBnpaMi AB i AC). To'IKaC Jie:>KHTh Mi:>K TO'IKRMHAi B, ocKiJihKH 3a yMOBOIO 3aAa'liBOHa HaJie:>KHTh Bi,l.\pi3Ky AB.

0Tme, TO'IKaA He Jie:>KHTh Mi:>K TO'IKaMHBi C, T06To TO'IKH

Bi C JiemaTh no OAHH 6iK Bi).\ TO'IKHA. ToMy niBnpHMi AB i AC 36iraIOThCH.

Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]