Геометрия7_9

8 KJiaC

112

68.

y TPHKYTHHKy O,ll;HH 3 KyTiB IIPH OCHOBi .z:i;opiBHIOG 45°'a BHCOTa

)l;iJIHTh OCHOBY

Ha 11aCTHHH 20 CM i 21 CM. 3Haii.z:i;iTh 6iJihIIIY

69.

6i11Hy CTOpOHY 1 (MaJI. 169).

y TPHKYTHHKY

O)l;Ha ia CTOpiH .z:i;opiBHIOG 1 M, a rrpHJierJii ,ll;O

Hei KYTH 30° i

45°. 3Haii.z:i;iTh ,ll;Bi iHIIIi CTOPOHH TpHKYTHHKa.

72. SIKHii a KyTiB 6iJihIIIHH -

a

'IH~. HK:rn;o:

! ;

1)

sin a=

+•

sin~=

~ ;

2)

sin a= ~ ,

sin~=

3

2

3)

cos a=

7 ,

cos~=

5 ;

4) cos a= 0,75,cos~= 0,74;

8

5

5)

tg a =

2,1,

tg ~ = 2,5;

6)

tg a= 3 ,

tg ~ = 2

?

KYJJHpHi rrpaMi x i y -- oci 1eoopaunar (MaJI.

170). Bich x (BoHa,

HK npaBHJIO, ropH30HTaJihHa) Ha3HBaGThCH BiCCIO aocu,uc, a BiCh

y - BiCCIO owunar. To11Ka ix nepeTHHY 0 -

no'laT01C 1COopau-

KomHiii To11:u;i

A

rrJio:rn;HHH

nocTaBHMO y

Bi.z:i;noBi.z:i;HiCTh rrapy

11HceJI -

1eoopaunaru

TO'l1CU -

a6c:u;Hcy (x)

i op.z:i;HHaTy (y) aa

TaKHM rrpaBHJIOM.

qepea

TO'IKY A

npoBe)l;eMo

rrps:My, rrapaJieJihHY oci op.z:i;HHaT

Ax HaJieJKHTh .z:i;o.z:i;aTHiii niBoci, i

Bi,ll;'GMHe, HK:rn;o

Ax HaJieJKHTh

Bi.z:i;'sMHiii

niBoci. KoJIH TO'IKa A

JieJKHTh Ha oci

op.z:i;HHaT y, TO

BBaJKaGMO,

:rn;o x .z:i;opiBHIOG HyJIIO.

§ 8. ,Il;eKaJ>TOBi KOOpp;HHSTH Ha IlJIO~Hi

115

0

3 a ,n; a 'Ia

(15). ,lJ;aHo

TPH

BepmHHH napaJieJiorpaMa

ABCD: A (1;

0), B (2; 3),

C (3;

2). 3Haii,n;iTL Koop,n;HHaTH

'!eTBepToisepmHHH D i TO'IKHnepeTHHY ,n;iaroHaJieii.

Po as'a a a H Ha. To'!KanepeTHHY ,n;iaroHaJieii e cepe,n;H-

HOIO KOmHoi a HHX. ToMy soHa e cepe,n;HHOIO si,n;piaKa AC,

a TOMY MaG KOOP,D;HHaTH

1+a

0+2

x = - 2 - =

2, y = - 2 - = 1.

. ·

BD

' MaGMO:

2 +x

2

a+y

1

.

3.

x =

2

,

Bl,D;pl3Ka

- 2 - =

, - 2 - =

Bl,ll;CH

y= -1.

Ai i A 2 qepea Koop,n;uHaTH Il;HX TO'IOK.

* X2

i Yt * y 2. IIpo-

PoarJiaHeMo cno'!aTKYsuna,n;oK, KOJIH Xi

Be,n;eMo qepea TO'IKHAi i A2 npaMi, napaJieJibHi

oci Koop,n;uHaT,

TO'IKaMHA ii A1 ,n;opiBHIOG lyi -

y2I,

a Bi,n;cTaHL Mim TO'IKaMHA

ii A 2 ,n;opiBHIOG lxi

-

x2I. 3a TeopeMOIO IIicparopa y npaMOKYTHOMY

TPHKYTHHKY AAiA 22MaGMO:

2

2

d

= (xi

- X2)

+

(yi -

Y2) ,

,n;e d -

si,n;cTaHb Mim TO'!KaMHAi i

A 2.

Xo11a cpopMyna (•) ,n;na Bi,n;cTaHi Mim TO'!KaMHBHBe,n;eHa y npu-

nyw;eHHi Xi *

X2, Yi

* y2, BOHa 3aJIHWa6TbC$1 npaBHJll>HOIO i

,ll;JISI

iHmux BHna,n;Kis. Cnpas,n;i, aKw;o xi = x 2,

yi * y2,

TO d ,n;opiBHIOG

lyi -

y2I. TaKHii

caMHii peayJILTaT ,n;icTaGMo i

aa

cpopMyJioIO

(•).

AHaJiori11Ho poarna,n;aeMo BHna,n;oK, KOJIH xi *

X2, yi = y 2• .HK~,o

Xi =

X2,

Yi =

Y2,

TO

TO'IKHAi i

A2

a6ira10TLCS1 i aa cpopMyJioIO (•) ,n;icTa-

y

HeMo d = 0.

(19).

A1

3 a ,n; a q a

3Haii,n;iTL

Ha oci x TO'!Ky,piBHOBi,n;,n;aJieHy

Bi,n;

TO'IOK(l; 2) i (2;

3).

P o a s 'a a a H H a.

Hexaii

(x;

0) -

myKaHa TO'!Ka. IIpH-

A

piBHSIBWH Bi,n;cTaHi Bi,n; Hei ,n;o

,n;aHHX TO'IOK,,n;icTaHeMO:

(x - 1)2 +

(0 - 2)2 =

x

= (x -

2)2

+ (0 -

3)2.

0

3Bi,D;CH 3HaXO,ll;HMO x =

4. 0Tme,

myKaHa TO'!Ka6 (4;

0).

Man. 174

8 KJiac

116

Bip; He'iAO n;eHTpa Ao AOPiBHIOe R. KBaApaT BiACTaHi BiA TO'IKHA

y

AO Ao AOPiBHI06 (x -

a)2

+ (y -

b)2.

TaKHM 'IHHOM,KOOPAHHaTH x, y KOmHo'i

TO'IKHA KoJia aaAOBOJihHSIIOTh piBHHHHH

(x - a)2

+ (y -

b)2 =

R 2•

(•)

HaBnaKH: 6yAh-RKa TO'IKaA, Koop-

AHHaTH SlKOl aaAOBOJlhHSllOTh piBHHHHH

(.), H&Jie:>KHTh KOJIY'OCKiJihKH BiACTaHh

BiA He'iAO TO'IKHAo AOpiBHI06 R. 3BiACH

0

x

BHilJIHB86, ~o piBH$1HHSl (.) cnpaBAi 6

piBHHHHHM KOJia a n;eHTpoM A o i

paAiY-

Ma;r.

175

coM R.

3ayeamHMO, ~o KOJIH n;eHTpOM KOJia

6 fiO'laTOKKOOPAHHaT, TO piBHHHHH KOJia Mae BHrJIHA:

a (30).

x2

+ y2= R2.

3 a A a q

.flKy reoMeTPH'IHY <l>irypy

3aA8HO

piB-

x 2

+ y 2 + ax + by + c =

a 2

+

bz

c > O?

HHHHHM

0, 4

4

-

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

IlepeTBopHMO

p;aHe

piBHHHHSI

TaK:

a z

bz

a2

bz

x 2 + ax + -

+ y 2 + by + -

= -

4

+ - - c,

4

4

4

( x + ~ y+ (y + ~ y= ( y~2 + b: - c y.

Ba1UIMO,

~o myKaHa <l>irypa -

KOJIO 3

n;eHTpoM

( - ~ ;

,ll;oBeAeMo, ~o 6yiJ-,,-sixa npsiMa

y iJexaJ11'06UX xoopiJuHarax

x, y MtJ.e pisHsiHHSi suiJy:

ax + by + c =

0, ( * ),

§ 8. ,!J;eKapTOBi KOOp,D;HHaTH Ha IIJIO~HHi

117

HiH Bi,n; TO'IKHnepeTHHY c 3 np.RMOIO h

h

piBHi

Bi,n;piaKH

CA i

i

CA2 (MaJI.

176).

Hexa:H ai,

bi -

Koop,n;HHaTH

TO'IKH

Ai i az, bz -- Koop,n;HHaTH TO'IKHAz. HK

Bi,n;oMo, 6y,n;b-HKa TO'IKaA(x; y) npnMoi

h piBHOBi,n;,n;aJieHa Bi,n;

TO'IOK Ai

i Az.

ToMy

Koop,n;HHaTH

'ii aa,n;oBOJibHHIOTb

piBHHHH.fl

- bi)2 =

(x - ai)2 + (y

= (x -

az)2 +

(y -

bz)2 •

(**)

HaBnaKH:

HK~o

Koop,n;HHaTH x, y

176

MaJI.

HKOi-He6y,n;b TO'IKHaa,n;oBOJibHHIOTb piB-

HHHHH (••),TO :a;n TO'IKapiBHOBi,n;,n;aJieHa

Bi,n; TO'IOK A 1 i Az,

To6To

HaJielKHTb npnMiH h. TaKHM 'IHHOM,

2 (a2 -

ai)x + 2 (b2 -

b1)y + (ar + br -

a§ - b§) = 0.

IIoaHa'IHBmH2 (a2 - ai) = a, 2 (b2 - bi)=

b,

ar + by - a§ -

-

b~

=

c, ,n;icTaHeMo piBHHHHH (•). TBep,n;meHHH

,n;oBe,n;eHo.

3 a ,n; a q a

35. CKJia,n;iTb piBH$1HHH

npnMoi, HKa npoxo-

,ll;HTb qepea TO'IKHA(-1; 1), B(l; 0).

Po a B'n a a H H n. HK Bi,n;oMo, piBHHHHa npnMoi Mae BH,n;

ax + by + c =

0. To'IKHA i B JiemaTb Ha npaMi:H, TOMY ix

KOOp,D;HHaTH 38,ll;OBOJibHHIOTb :a;e piBHHHH.fl.

IIi,n;cTaBHBmH Koop,n;HH&TH TO'IOKA i

By piBHHHHH np.R-

MOi, ,n;icTaHeMo:

-a + b + c = 0, a + c = 0.

3 D;HX piBHHHb MOlKHa BHP83HTH ,ll;B& KOeci>i:a;ieHTH, HanpHK-

Jia,n; a i

b, qepea TpeTiii: a= -c, b =

-2c. IIi,n;cTaBJIHIO'IH

:u;i

3Ha'leHH.fla

i b y

piBHHHH.fl IIp.RMOi,

,n;icTaHeMo:

-ex - 2cy + c =

0.

Ha c MOlKHa CKopOTHTH. To,n;i MaTHMeMo:

- x - 2y +

1 =

0.

D;e i e piBHHHHH myKaHoi npHMoi.

76. KOOP,IJ;HHATH TO'IKH IIEPETHHY IIPHMHX

Hexa:H ,n;aHo piBHHHHH ,n;Box np.RMHX

ax+ by+ c =

0,

aiX + biy + C1 = 0.

3Ha:H,n;eMO KOOp,ll;HH&TH ix TO'IKHnepeTHHy.

OcKiJibKH TO'IKanepeTHHY (x; y) H&JiemHTb KOmHiii a npHMHX,

TO

ii

KOOp,ll;HH&TH aa,n;OBOJibHHIOT& i

nepme

i

,n;pyre piBH$1HHH.