Chuyen de Phep Bien Hinh Trong Mp

7/27/2019 Chuyen de Phep Bien Hinh Trong Mp

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- 2 -

2 2 2 22 1 2 1 2 1 2 1

1 2

Ta co : MN = (x x ) (y y ) , M N = 9(x x ) (y y )

Neu x x th M N MN . Vay : f khong phai la phep di hnh .(V co 1 so iem f khong bao toan khoang cach) .

4 Trong mpOxy cho 2 phep bien hnh :

a) f : M(x;y) M = f(M) = (y ; x-2) b) g : M(x;y) M = g(M) = ( 2x ; y+1) .

Phep bien hnh nao tren ay la phep di hnh ?HD :

I I

1 2a) f la phep di hnh b) g khong phai la phep di hnh ( v x x th M N MN )


a) f : M(x;y) M = f(M) = (y + 1 ; x) b)I

1

g : M(x;y) M = g(M) = ( x ; 3y ) .

Phep bien hnh nao tren ay la phep di hnh ?Giai :a) f la phep di hnh b) g khong phai la phep di hnh ( v y y

I

2 th M N MN )

6 Trong mpOxy cho phep bien hnh f : M(x;y) M = f(M) = ( 2x ;y 1) . Tm anh cua ng

thang ( ) : x 3y 2 = 0 qua phep bien hnh f .Giai :Cach 1: Dung bieu thc toa o

I

xx = 2x xTa co f : M(x;y) M = f(M) = 2y y 1

y y 1

xV M(x;y) ( ) ( ) 3(y 1) 2 0 x 6y 2 0 M (x ;y ) ( ) : x 6y 2 0

2Cach 2 : Lay 2 iem ba t k M,N ( ) : M N .

+ M

I

( ) : M(2;0) M f(M) ( 4;1)

+ N ( ) : N( 1; 1) N f(N) (2;0)

I

I

Qua M ( 4;1) x+ 4 y 1( ) (M N ) : PTCtac ( ) : PTTQ ( ) : x 6y 2 0

6 1VTCP : M N (6; 1)

2 2

7 Trong mpOxy cho phep bien hnh f : M(x;y) M = f(M) = (x 3;y 1) .a) CMR f la phep di hnh .

b) Tm anh cua ng tron (C) : (x + 1) + (y 2) = 4 . (C ) : (x

I

I 2 22) + (y 3) = 4

8 Trong mpOxy cho phep bien hnh f : M(x;y) M = f(M) = (x 3;y 1) .

a) CMR f la phep di hnh .b) Tm anh cua ng thang ( ) : x+ 2y 5 = 0 .

c) Tm anh cua ng tron (C) : (x

I

2 2+ 1) + (y 2) = 2 .

2 2x y

d ) Tm anh cua elip (E) : + = 1 .3 2

1 1 2 2

1 1 1 1

2 2 2 2

Giai : a) Lay hai iem bat k M(x ;y ),N(x ;y )

Khi o f : M(x ;y ) M = f(M) = (x 3; y 1) .

f : N(x ;y ) N = f(N) = (x 3; y 1)

Ta co : M N = (

I

I

2 22 1 2 1x x ) (y y ) = MN

Vay : f la phep di hnh .


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- 3 -

b) Cach 1: Dung bieu thc toa ox = x 3 x x 3Ta co f : M(x;y) M = f(M) =y y 1 y y 1

V M(x;y) ( ) (x 3) 2(y 1) 5 0 x 2y 4 0 M (x ;y ) (

I

) : x 2y 4 0

Cach 2: Lay 2 iem bat k M,N ( ) : M N .

+ M ( ) : M(5 ;0) M f(M) (2;1)

+ N ( ) : N(3 ; 1) N f(N) (0;2)

I

I

Qua M (2;1) x 2 y 1( ) (M N ) : PTCtac ( ) : PTTQ( ) : x 2y 4 0

2 1VTCP : M N ( 2;1)

Cach 3: V f la phep di hnh nen f bien ng thang ( ) thanh ng thang (

) // ( ) .

+ Lay M ( ) : M(5 ;0) M f(M) (2;1)+ V ( ) // ( ) ( ) : x + 2y m = 0 (m 5) . Do : ( ) M (2;1) m = 4 ( ) : x 2y 4 0

c) Cach 1: Dung bieu thc toa o

I

2 2 2 2

x = x 3 x x 3Ta co f : M(x;y) M = f(M) =y y 1 y y 1

V M(x;y) (C) : (x + 1) + (y 2) = 2 (x 4) (y 3) 2M(x;y)

I

2 2

f 2 2

(C ) : (x 4) (y 3) 2

+ Tam I( 1;2) + Tam I = f [ I( 1;2)] ( 4;3)Cach 2 : (C) (C ) (C ) : (x 4) (y 3) 2

BK : R = 2 BK : R = R = 2

d) Dung bieu thc toa ox = x 3 x x 3Ta co f : M(x;y) M = f(M) =y y 1 y y 1

I

2 2 2 2 2 2x y (x + 3) (y 1) (x + 3) (y 1)

V M(x;y) (E) : + = 1 + = 1 M (x ;y ) (E ) : + = 1

3 2 3 2 3 2

9 Trong mpOxy cho phep bien hnh f : M(x;y) M = f(M) = (x 1;y 2) .a) CMR f la phep di hnh .b) Tm anh cua ng thang ( ) : x 2y 3

I

2 2

2

2 2 2

= 0.

c) Tm anh cua ng tron (C) : (x + 3) + (y 1) = 2 .

d) Tm anh cua parabol (P) : y = 4x .

S : b) x 2y 2 = 0 c) (x + 2) + (y 1) = 2 d) (y + 2) = 4(x 1)

10 Trong mpOxy cho phep bien hnh f : M(x;y) M = f(M) = ( x ;y) . Khang nh nao sau aysai ?

I

A. f la 1 phep di hnh B. Neu A(0 ; a) th f(A) = AC. M va f(M) oi xng nhau qua truc hoanh D. f [M(2;3)] ng thang 2x + y + 1 = 0

S : Chon C . V M va f(M) oi xng nhau qua truc tung C sai .

1 1 2 2

1 2


f : M(x;y) M = f (M) = (x + 2 ; y 4) ; f : M(x;y) M = f (M) = ( x ; y) .

Tm toa o anh cua A(4; 1) qua f roi f , ngha la tI I

1 22 1

f f

m f [f (A)] .

S : A(4; 1) A (6; 5) A ( 6 ; 5 ) .I I

x

11 Trong mpOxy cho phep bien hnh f : M(x;y) M = f(M) = ( ; 3y) . Khang nh nao sau ay sai ?2A. f (O) = O (O la iem bat bien) B. Anh cua A Ox th

I

anh A = f(A) Ox .C. Anh cua B Oy th anh B = f(B) Oy . D. M = f[M(2 ; 3)] = (1; 9)

S : Chon D . V M= f[M(2 ; 3)] = (1;9)


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Vn 2 : PHP TNH TINA.KIN THC C BN

1/ N: Php tnh tin theo vct u l mt php di hnh bin im M thnh im M sao cho uMM .

K hieu : T hay T .Khi o : T (M) M MM uu uPhep tnh tien hoan toan c xac nh khi biet vect tnh tien cua no .

Neu T (M) M , M th T la phep ong nhat .o o

2/ Biu thc ta : Cho u = (a;b) v php tnh tin Tu .

x = x + aM(x;y) M =T (M) (x ; y ) thu y = y + bI

3/ Tnh cht:L : Phep tnh tien bao toan khoang cach gia hai iem bat k .HQ :

1. Bao toan tnh thang hang va th t cua cac iem tng ng .2. Bien mot tia thanh tia .3. Bao toan tnh thang hang va th t cua cac iem tng ng .

5. Bien mot oan thang thanh oan thang bang no .6. Bien mot ng thang thanh mot ng thang song song hoac trung vi ng thang a cho .

Bien7. tam giac thanh tam giac bang no . (Trc tam trc tam , trong tam trong tam )I I

8. ng tron thanh ng tron bang no .

(Tam bien thanh tam : I I , R = R )I

PHNG PHP TM NH CA MT IM

x = x + aM(x;y) M =T (M) (x ; y ) thu y = y + bI

PHNG PHP TM NH CA MT HNH (H) .Cch 1: Dng tnh cht (cng phng ca ng thng, bn knh ng trn: khng i)

1/ Ly M (H) M (H )I

2/ (H) ng thang (H ) ng thang cung phng

Tam I Tam I(H) (C) (H ) (C ) (can tm I ) .+ bk : R + bk : R = R

II

Cach 2 : Dung bieu thc toa o .Tm x theo x , tm y theo y roi thay vao bieu thc toa o .

Cach 3 : Lay hai iem phan biet : M, N (H) M , N (H )I

B. BI TP

1 Trong mpOxy . Tm anh cua M cua iem M(3; 2) qua phep tnh tien theo vect u = (2;1) .Giai

x 3 2 x 5Theo nh ngha ta co : M = T (M) MM u (x 3;y 2) (2;1)u y 2 1 y 1

M (5; 1)

2 Tm anh cac iem ch ra qua phep tnh tien theo vect u :

a) A( 1;1) , u = (3;1)

A (2;3)b) B(2;1) , u = ( 3;2)

B ( 1;3)

c) C(3; 2) , u = ( 1;3) C (2;1)


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3 Trong mpOxy . Tm anh A ,B lan lt cua iem A(2;3), B(1;1) qua phep tnh tien theo vect u = (3;1) .

Tnh o dai AB , A B .Giai

Ta co : A = T (A) (5;4) , B = T (B)u u

1 2

1 2

(4;2) , AB = |AB| 5 , A B = |A B | 5 .

4 Cho 2 vect u ;u . Ga s M T (M),M T (M ). Tm v e M T (M) .1 2 1 u 2 u 1 2 vGiai

Theo e : M T (M) MM u , M T (M ) M M1 u 1 1 2 u 1 1 2

u .2

Neu : M T (M) MM v v MM MM M M u + u .Vay : v u + u2 v 2 2 1 1 2 1 2 1 2

5 ng thang cat Ox tai A( 1;0) , cat Oy tai B(0;2) . Hay viet phng trnh ng thang la anhcua qua phep tnh tien theo vect u = (2; 1) .

Giai V : A T (A) (1; 1) , B T (B) (2;1) .u uqua A (1; 1) x 1 tMat khac : T ( ) i qua A ,B . Do o : ptts :u y 1 2tVTCP : A B = (1;2)

6 ng thang cat Ox tai A(1;0) , cat Oy tai B(0;3) . Hay viet phng trnh ng thang la anhcua qua phep tnh tien theo vect u = ( 1; 2) .GiaiV : A T (A) (0; 2) ,u

B T (B) ( 1;1) .u

qua A (0; 2) x tMat khac : T ( ) i qua A ,B . Do o : ptts :u y 2 3tVTCP : A B = ( 1;3)

7 Tng t : a) : x 2y 4 = 0 , u = (0 ; 3) : x 2y 2 0b) : 3x y 3 = 0 , u = ( 1 ; 2) : 3x y 2 0

2 28 Tm anh cua ng tron (C) : (x + 1) (y 2) 4 qua phep tnh tien theo vect u = (1; 3) .Giai

x = x + 1 x = x 1Bieu thc toa o cua phep tnh tien T la :u y = y 3 y = y + 3

V

2 2 2 2 2 2 : M(x;y) (C) : (x + 1) (y 2) 4 x (y 1) 4 M (x ;y ) (C ) : x (y 1) 42 2Vay : Anh cua (C) la (C ) : x (y 1) 4

9 Trong mpOxy cho phep bien hnh f : M(x;y) M = f(M) = (x 1;y 2) .a) CMR f la phep di hnh .

b) Tm anh cua ng thang ( ) : x 2y 3

I

2 2

2

2 2 2

= 0.c) Tm anh cua ng tron (C) : (x + 3) + (y 1) = 2 .

d) Tm anh cua parabol (P) : y = 4x .

S : b) x 2y 2 = 0 c) (x + 2) + (y 1) = 2 d) (y + 2) = 4(x

1)

10 Trong mpOxy cho phep bien hnh f : M(x;y) M = f(M) = ( x ;y) . Khang nh nao sau aysai ?

A. f la 1 phep di hnh B.

I

Neu A(0 ; a) th f(A) = A

C. M va f(M) oi xng nhau qua truc hoanh D. f [M(2;3)] ng thang 2x + y + 1 = 0S : Chon C . V M va f(M) oi xng nhau qua t ruc tung C sai .

2 29 Tm anh cua ng tron (C) : (x 3) (y 2) 1 qua phep tnh tien theo vect u =( 2;4) .x = x 2 x = x + 2Giai : Bieu thc toa o cua phep tnh tien T la :u y = y 4 y = y 4


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2 2 2 2 2 2V : M(x;y) (C) : (x 3) (y 2) 1 (x 1) (y 2) 1 M (x ;y ) (C ) : (x 1) (y 2) 12 2Vay : Anh cua (C) la (C ) : (x 1) (y 2) 1

2 2 2 2BT Tng t : a) (C) : (x 2) (y 3) 1, u = (3;1) (C ) : (x 1) (y 2) 12 2

b) (C) : x y 2x 4y 4 0, u = ( 2;3) (C ) 2 2: x y 2x 2y 7 0

10 Trong he truc toa o Oxy , xac nh toa o cac nh C va D cua hnh bnh hanh ABCD biet nh

A( 2;0), nh B( 1;0) va giao iem cac ng cheo la I(1;2) .Giai

Goi C(x;y) .Ta

co : IC (x 1;y 2),AI (3;2),BI (2; 1)V I la trung iem cua AC nen :

x 1 3 x 4C = T (I) IC AI C(4;4)

AI y 2 2 y 4

V I la trung iem cua AC nen :

D = T (I) IDBI

x 1 2 x 3D DBI D(3;4)y 2 2 y 4D D

Bai tap tng t : A( 1;0),B(0;4),I(1;1) C(3;2),D(2; 2) . 11 Cho 2 ng thng song song nhau dv d . Hy ch ra mt php tnh tin bin d thnh d . Hi c bao

nhiu php tnh tin nh th?

Gia i : Chon 2 iem co nh A d , A dLay iem tuy y M d . Ga s : M = T (M) MM AB

AB

MA M B M B/ /MA M d d = T (d)AB

Nhan xet : Co vo so phep tnh

tien bien d thanh d .

12 Cho 2 ng tron (I,R) va (I ,R ) .Hay ch ra mot phep tnh tien bien (I,R) thanh (I ,R ) .Giai : Lay iem M tuy y tren (I,R) . Ga s : M = T (M) M

II

M II

IM I M I M IM R M (I ,R ) (I ,R ) = T [(I,R)]II

13 Cho hnh bnh hanh ABCD , hai nh A,B co nh , tam I thay oi di ongtren ng tron (C) .Tm quy tch trung iem M cua canh BC.

Giai

Goi J la trung iem canh AB . Khi o d e thay J co nh va IM JB .Vay M la anh cua I qua phep tnh tien T . Suy ra : Quy tch cua M la

JB

anh cua ng tron (C) trong phep tnh tien theo vect JB

214 Trong he truc toa o Oxy , cho parabol (P) : y = ax . Goi T la phep tnh tien theo vect u = (m,n)

va (P ) la anh cua (P) qua phep tnh tien o . Hay viet phng trnh cua

u

(P ) .Giai :

TM(x;y) M (x ;y ) , ta co : MM = u , vi MM = (x x ; y y)

x x = m x = x mV MM = uy y = n y = y n

2 2Ma : M(x;y) (P) : y ax y n = a(x m) y =

I

2 2a(x m) n M (x ;y ) (P ) : y = a(x m) n

2 2 2Vay : Anh cua (P) qua phep tnh tien T la (P ) : y = a(x m) n y = ax 2amx am n .u15 Cho t : 6x + 2y 1= 0 . Tm vect u 0 e = T ( ) .uGi

ai : VTCP cua la a = (2; 6) . e : = T ( ) u cung phng a . Khi o : a = (2; 6) 2(1; 3)u

chon u = (1; 3) .16 Trong he truc toa o Oxy , cho 2 iem A( 5;2) , C( 1;0) . Biet : B = T (A) , C = T (B) . Tm u va vu ve co the thc hien phep bien oi A thanh C ?

Giai


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Tu+ v

u vT T

A( 5;2) B C( 1;0)I I . Ta co : AB u,BC v AC AB BC u v (4; 2)

u v

17 Trong he truc toa o Oxy , cho 3 iem K(1;2) , M(3; 1),N(2; 3) va 2 vectu = (2;3) ,v = ( 1;2) .Tm anh cua K,M,N qua phep tnh tien T roi T .u v

T THD : Ga s : A(x;y) BI I

C(x ;y ) . Ta co : AB u,BC v AC AB BC u v (1;5)x 1 1 x 2

Do o : K =T (K) KK (1;5) K (2;7) .u v y 2 5 y 7Tng t : M (4;4) , N(3;2) .

18 Trong he truc toa o Oxy , cho ABC : A(3;0) , B( 2;4) , C( 4;5) . G la trong tam ABC va pheptnh tien theo vect u 0 bien A thanh G . Tm G = T (G) .u

u u

GiaiT T

A(3;0) G( 1;3) G (x ;y )

x 1 4 x 5V AG ( 4;3) u . Theo e : GG u G ( 5;6).y 3 3 y 62 219 Trong mat phang Oxy , cho 2 ng tron (C) : (x 1) (y 3) 2,(

I I

2 2C ) : x y 10x 4y 25 0.

Co hay khong phep tnh tien vect u bien (C) thanh (C ) .HD : (C) co tam I(1; 3), ban knh R = 2 ; (C ) co tam I (5; 2), ban knh R = 2 .

Ta thay : R =

R = 2 nen co phep tnh tien theo vect u = (4;1) bien (C) thanh (C ) .

20 Trong he truc toa o Oxy , cho hnh bnh hanh OABC vi A( 2;1) va B :2x y 5 = 0 . Tm taphp nh C ?

Giai

u

V OABC la hnh bnh hanh nen : BC AO (2; 1) C T (B) vi u = (2; 1)uT x x 2 x x 2

B(x;y) C(x ;y ) . Do : BC uy y 1 y y 1

B(x;y) 2x y 5 = 0 2x y 10 = 0 C(x ;

I

y ) : 2x y 10 = 0

21 Cho ABC . Goi A ,B ,C lan lt la trung iem cac canh BC,CA,AB. Goi O ,O ,O va I ,I ,I1 1 1 1 2 3 1 2 3tng ng la cac tam ng tron ngoai tiep va cac tam ng tro

1AB

2

n noi tiep cua ba tam giac AB C ,1 1BC A , va CA B . Chng minh rang : O O O I I I .1 1 1 1 1 2 3 1 2 3

HD :

Xet phep tnh tien : T bien A C,C B,B A .1 1 1 1AB2

T

AB C C BA ;O1 1 1 1

I I I

I

1 1AB AB

2 2

T T

O ;I I .1 2 1 2

O O I I O O I I .1 2 1 2 1 2 1 2Ly luan tng t : Xe t cac phep tnh tien T ,T suy ra :1 1

BC CA2 2

O O I I va O O I2 3 2 3 3 1 3

I I

I O O I I ,O O I I O O O I I I (c.c.c).1 2 3 2 3 3 1 3 1 1 2 3 1 2 3


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BC

22 Trong t giac ABCD co AB = 6 3cm ,CD 12cm , A 60 ,B 150 va D 90 .Tnh o dai cac canh BC va DA .

HD :

TXet : A M AM BC.Ta co : ABCM la hnh bnh hanh va BCM 3I 0 (v B 150 )

oLai co : BCD 360 (90 60 150 ) 60 MCD 30 .

nh ly ham cos trong MCD : 32 2 2 2 2MD MC DC 2MC.DC.cos30 (6 3) (12) 2.6 3.12. 362

MD = 6cm .

1Ta co : MD = CD va MC = MD 3 MDC la tam giac

2

eu

MCD la na tam giac eu DMC 90 va MDA 30 .

Vay : MDA MAD MAB 30 AMD la tam giac can tai M .

6 3

Dng MK AD K la trung iem cua AD KD=MDcos30 cm AD 6 3cm2

Tom lai : BC = AM = MD = 6cm , AD = AB = 6 3cm

Vn 3: PHP I XNG TRCA . KIN THC C BN

1/ N1:im M gi l i xng vi im Mqua ng thng a nu a l ng trung trc ca on MM Phep oi xng qua ng thang con goi la phep oi xng truc . ng thang a goi la truc oi xng.

N2 : Phep oi xng qua ng thang a la phep bien hnh bien mo

a o o o

i iem M thanh iem M oi xngvi M qua ng thang a .

K hieu : (M) M M M M M , vi M la hnh chieu cua M tren ng thang a .

Khi : aNeu M a th (M) M : xem M la oi xng vi chnh no qua a . ( M con goi la iem bat ong )

aM a th (M) M a la ng trung trc cua MM

a a (M) M th (M ) M

a a (H) H th (H ) H , H la anh cua hnh H .

dN : d la truc o i xng cua hnh H (H) H .Phep oi xng truc hoan toan xac nh khi biet truc oi xng cua no .

Chu y : Mot hnh co the khong co truc oi xng ,co the co mot hay nhieu truc oi xng .

2/ Biu thc ta : dM(x;y) M (M) (x ;y )I

x = x x = x d Ox : d Oy :y = y y = y

3/ L:Php i xng trc l mt php di hnh.

1.Phep oi xng truc bien ba iem thang hang thanh ba iem thang hang va bao toan th t cua caciem tng ng .

2. ng thang thanh ng thang .3.

HQ :

Tia thanh tia .

4. oan thang thanh oan thang bang no .5. Tam giac thanh tam giac bang no . (Trc tam trc tam , trong tam trong tam )

6. ng tron thanh ng

I I

tron bang no . (Tam bien thanh tam : I I , R = R )7. Goc thanh goc bang no .

I


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aPP : Tm anh M = (M)

1. (d) M , d a2. H = d a

3. H la trung iem cua MM M ?

a

a

PP : Tm anh cua ng thang : = ( )TH1:( ) // (a)

1. Lay A,B ( ) : A B2. Tm anh A = (A)

3. A , // (a)

a

TH2 : // a

1. Tm K = a2. Lay P : P K .Tm Q = (P)3. (KQ)

PP: minTm M ( ) : (MA + MB) .

min

min

Tm M ( ) : (MA+ MB)

Loai 1 : A, B nam cung pha oi vi ( ) :1) goi A la oi xng cua A qua ( )2) M ( ), th MA + MB MA + MB A BDo o: (MA+MB) = A B M = (A B) ( )

min

Loai 2 : A, B nam khac pha oi vi ( ) :M ( ), th MA + MB AB

Ta co: (MA+MB) = AB M = (AB) ( )

B . BI TP

OyOx

1 Trong mpOxy . Tm anh cua M(2;1) oi xng qua Ox , roi oi xng qua Oy .

HD : M(2;1) M (2; 1) M ( 2; 1)

2 Trong mpOxy . Tm anh cua M(a;b) oi xng qua Oy , roi oi x

I I

Oy Ox

a b

a b

ng qua Ox .

HD : M(a;b) M ( a;b) M ( a; b)

3 Cho 2 ng thang (a) : x 2 = 0 , (b) : y + 1 = 0 va iem M( 1;2) . Tm : M M M .

HD : M( 1;2) M (5;2)

I I

I I

I I

a b

a bt(m;y) t(

M (5; 4) [ ve hnh ] .

4 Cho 2 ng thang (a) : x m = 0 (m > 0) , (b) : y + n = 0 (n > 0).

Tm M : M(x;y) M (x ;y ) M (x ;y ).

x 2m xHD : M(x;y) My y

I I

2m x; n)x 2m xMy 2n y

5 Cho iem M( 1;2) va ng thang (a) : x + 2y + 2 = 0 .HD : (d) : 2x y + 4 = 0 , H = d a H( 2;0) , H la trung iem cua MM M ( 3; 2)

6 Cho iem M( 4;

a

a

1) va ng thang (a) : x + y = 0 . M = (M) ( 1;4)

7 Cho 2 ng thang ( ) : 4x y + 9 = 0 , (a) : x y + 3 = 0 . Tm anh = ( ) .HD :

4 1V

1

a

cat a K a K( 2;1)1

M( 1;5) d M, a d : x y 4 0 H(1/ 2;7 / 2) : tiem cua MM M (M) (2;2)KM : x 4y + 6 = 0


10/35

- 10 -

a

a

a

8 Tm b = (Ox) vi ng thang (a) : x + 3y + 3 = 0 .HD : a Ox = K( 3;0) .

3 9M O(0;0) Ox : M = (M) = ( ; ) .

5 5b KM : 3x + 4y 9 = 0 .

9 Tm b = (Ox) vi ng thang (a) : x + 3y 3 = 0 .

HD : a Ox = K(3;0) .

P O(0;0) Ox .

+ Qua O(0;0): 3x y 0

+ a

3 9 3 9E = a E( ; ) la trung iem OQ Q( ; ) .

10 10 5 5b KQ : 3x + 4y 9 = 0 .

1

Ox

Ox

0 Tm b = (a) vi ng thang (a) : x + 3y 3 = 0 .Giai :Cach 1: Dung bieu thc toa o (rat hay)

Cach 2 : K= a Ox K(3;0)P(0;1) a Q = (P) = (0; 1) b KQ : x 3y 3 = 0 .

a11 Cho 2 ng thang ( ) : x 2y + 2 = 0 , (a) : x 2y 3 = 0 . Tm anh = ( ) .PP : / /a

Cach 1 : Tm A,B A ,B A BCach 2 : Tm A A / / , A

a

2 2a

2 2

Giai: A(0;1) A (A) (2; 3)A , / / : x 2y 8 0

12 Cho ng tron (C) : (x+3) (y 2) 1 , ng thang (a) : 3x y + 1= 0 . Tm (C ) = [(C)]

HD : (C ) : (x 3) y 1 .

Ox

13 Trong mpOxy cho ABC : A( 1;6),B(0;1) va C(1;6) . Khang nh nao sau ay sai ?A. ABC can B B. ABC co 1 truc oi xngC. ABC ( ABC) OyD. Trong tam : G = (G)

HD : Chon D

2 214 Trong mpOxy cho iem M( 3;2), ng thang ( ) : x + 3y 8 = 0, ng tron (C) : (x+3) (y 2) 4.Tm anh cua M, ( ) va (C) qua phep oi xng truc (a) : x 2y + 2 = 0 .

Giai : Goi M ,

( ) va (C ) la anh cua M, ( ) va (C) qua phep oi xng truc a .Qua M( 3;2)

a) Tm anh M : Goi ng thang (d) : a+ (d) (a) (d) : 2x y + m = 0 . V (d) M( 3;2) m = 4 (d) : 2x y 4 = 0

H M M

H M M

MM

MM

1x (x x )

2+ H = (d) (a) H( 2;0) H la trung iem cua M,M H1

y (y y )2

12 ( 3 x ) x 12 M ( 1; 2)

1 y 20 (2 y )

2

b) Tm anh ( ) :1 3

V ( ) cat (a1 2

) K= ( ) (a)

x + 3y 8 = 0Toa o cua K la nghiem cua he : K(2;2)x 2y + 2 = 0


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- 11 -

aLay P K Q = [P( 1;3)] = (1; 1) . ( Lam tng t nh cau a) )

Qua P( 1;3)Goi ng thang (b) :a

E P Q Q

E P Q Q

+ (b) (a) (b) : 2x y + m = 0 . V (b) P( 1;3) m = 1 (b) : 2x y 1 = 0+ E = (b) (a) E(0;1) E la trung iem cua P,Q

1 1x (x x ) 0 ( 1 x ) x

2 2E 1 1

y (y y ) 1 (3 y )2 2

Q

Q

1

Q(1; 1)y 1

Qua K(2;2) x 2 y 2+ ( ) (KQ) : ( ) : 3x y 4 0

1 3VTCP :KQ ( 1; 3) (1;3)

a a

c) + Tm anh cua tam I( 3;2) nh cau a) .Tam I Tam I+ V phep oi xng truc la phep di hnh nen (C): (C ) : .Tm I IR 2 R R 2

+ Tam I( 3;2)Vay : (C)BK :

I I

a a

2 2

2 2+ Tam I = [I( 3; 2)] ( ; )(C ) 5 5R = 2

BK : R = R = 2

2 2(C ) : (x ) (y ) 4

5 5

I

2 215 Trong mpOxy cho iem M(3; 5), ng thang ( ) : 3x + 2y 6 = 0, ng tron (C) : (x+1) (y 2) 9.Tm anh cua M, ( ) va (C) qua phep oi xng truc (a) : 2x y + 1 = 0 .

HD :

a) M(3; 5) I

a

a

33 1 9 13M ( ; ),(d) : x 2y 7 0,tiem H( ; )

5 5 5 54 15

b) + K= (a) K( ; )7 7

+ P ( ) : P(2;0) K , Q = [P(2;0)] = ( 2;2) ( ) (KQ) : x 18y 38 0

c) + I(1; 2) 2 2a 9 8 9 8I ( ; ) , R = R = 3 (C ) : (x + ) (y ) 9

5 5 5 5I

2 2

Ox

16 Cho iem M(2; 3), ng thang ( ) : 2x + y 4 = 0, ng tron (C) : x y 2x 4y 2 0.Tm anh cua M, ( ) va (C) qua phep oi xng qua Ox .

x xHD : Ta co : M(x;y) M (y y

Ox

x x1) (2)

y y

Thay vao (2) : M(2; 3) M (2;3)

2 2 2 2

2 2 2 2

M(x;y) ( ) 2x y 4 = 0 M (x ;y ) ( ) : 2x y 4 = 0 .

M(x;y) (C) : x y 2x 4y 2 0 x y 2x 4y 2 0(x 1) (y 2) 3 M (x ;y ) (C ) : (x 1) (y 2) 3

Ox

Ox

17 Trong mpOxy cho ng thang (a) : 2x y+3 = 0 . Tm anh cua a qua .

x x x xGiai : Ta co : M(x;y) My y y y

V M(x;y) (a) : 2x y+3 = 0 2(x ) ( y )+3 = 0 2x y +3 = 0 M (

I

Oy

x ;y ) (a ) : 2x y + 3 = 0

Vay : (a) (a ) : 2x y + 3 = 0I

2 2Oy

Oy

2 2 2 2 2

18 Trong mpOxy cho ng tron (C) : x y 4y 5 = 0 . Tm anh cua a qua .

x x x x

Giai : Ta co : M(x;y) M y y y yV M(x;y) (C) : x y 4y 5 = 0 ( x ) y 4(y ) 5 = 0 x

I

2

2 2

Oy 2 2

y 4y 5 = 0

M (x ;y ) (C ) : x y 4y 5 = 0

Vay : (C) (C ) : x y 4y 5 = 0I


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- 12 -

2 2

a

a

19 Trong mpOxy cho thang (a) : 2x y 3 = 0 , ( ) : x 3y 11 = 0 , (C) : x y 10x 4y 27 = 0 .a) Viet bieu thc giai tch cua phep oi xng truc .b) Tm anh cua iem M(4; 1) qua .

a a

2 2

a

c) Tm anh : ( ) = ( ),(C ) (C) .Giai

a) Tong quat (a) : Ax + By + C=0 , A B 0

Goi M(x;y) M (x ; y ) , ta co : MM (x x; y y) cung phng VTPT n = (A;B) MM tnx

I

2 2

x x y yx At x x At( t ) . Go i I la trung iem cua MM nen I( ; ) (a)

y y Bt y y Bt 2 2

x x y y x x At y y BtA( ) B( ) C 0 A( ) B( ) C 0

2 2 2 22(Ax + By + C)

(A B )t 2(Ax + By + C) tA

2 2

2 2 2 2

a

B

2A(Ax + By + C) 2B(Ax + By + C)x x ;y y

A B A B

4(2x y 3) 3 4 12x x x x y

5 5 5 5Ap dung ket qua tren ta co :2(2x y 3) 4 3 6

y y y y y5 5 5 5

4 7b) M(4; 1) M ( ;

5I

a

2 2a

)5

c) : 3x y 17 0

d) (C) (C ) : (x 1) (y 4) 2

I

I

20 Trong mpOxy cho ng thang ( ) : x 5y 7 = 0 va ( ) : 5x y 13 = 0 . Tm phep oi xng quatruc bien ( ) thanh ( ) .

Giai1 5

V ( ) va ( ) cat nhau . Do o truc oi xng (a) cua phep oi xng bien ( ) thanh ( ) chnh5 1

la ng phan giac cua goc tao bi ( ) va ( ) .

1

2

1 2

x y 5 0 (a )| x 5y 7 | | 5x y 13|T o suy ra (a) :

x y 1 0 (a )1 25 25 + 1Vay co 2 phep oi xng qua cac truc ( ) : x y 5 0 , ( ) : x y 1 0

a21 Qua phep oi xng truc :1. Nhng tam giac nao bien thanh chnh no ?2. Nhng ng tron nao bien thanh chnh no ?

2 2

HD :

1. Tam giac co 1 nh truc a , hai nh con lai oi xng qua truc a .2. ng tron co tam a .

22 Tm anh cua ng tron (C) : (x 1) (y 2) 4 qua phep oi xng truc Oy.

2 2PP : Dung bieu thc toa o S : (C ) : (x 1) (y 2) 4

23 Hai ABC va A B C cung nam trong mat phang toa o va oi xng nhau qua truc Oy .

Biet A( 1;5),B( 4;6),C (3;1) . Hay

tm toa o cac nh A , B va C .S : A (1;5), B (4;6) va C( 3;1)

24 Xet cac hnh vuong , ngu giac eu va luc giac eu . Cho biet so truc oi xng tng ng cua moiloai a giac eu o va ch ra cach ve cac truc oi xng o .


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- 13 -

S :Hnh vuong co 4 truc oi xng , o la cac ng thang i qua 2 nh oi dien va cac ng thangi qua trung iem cua cac cap canh oi dien .

Ngu giac eu co 5 truc oi xng ,o la cac ng thang i qua nh oi dien va tam cua ngu giac eu .Luc giac eu co 6 truc oi xng , o la cac ng thang i qua 2 nh oi dien va cac ng thang iqua trung iem cua cac cap canh oi dien .

d25 Goi d la phan giac trong tai A cua ABC , B la anh cua B qua phep oi xng truc . Khang nh

nao sau ay sai ?A. Neu AB < AC th B tren canh AC .

d

B. B la trung iem canh AC .C. Neu AB = AC th B C .D. Neu B la trung iem canh AC th AC = 2AB .

S : Neu B = (B) th B AC .

A ung . V AB < AC ma AB = AB nen AB < AC B tren canh AC .1

B sai . V gia thiet bai toan khong u khang nh AB = AC.2

C ung . V AB = AB ma AB = AC nen AB = AC B C .

a b a b

D ung . V Neu B la trung iem canh AC th AC=2AB ma AB =AB nen AC=2AB .

26 Cho 2 ng thang a va b cat nhau tai O . Xet 2 phep oi xng truc va :

A B CI I

. Khang nh nao sau ay khong sai ?A. A,B,C ng tron (O, R = OC) .B. T giac OABC noi tiep .C. ABC can BD. ABC vuong B

1 2HD : A. Khong sai . V d la trung trc cua AB OA = OB , d la trung trc

cua BC OB = OC OA = OB = OC A,B,C ng tron (O, R = OC) .Cac cau B,C,D co the sai .

27 Cho ABC co hai truc oi xng . Khang nh nao sau ay ung ?A. ABC la vuong B. ABC la vuong can C. ABC la eu D. ABC la can .

HD: Ga s ABC co 2truc oi xng la AC va BCAB = AC AB AB BC ABC eu .BC = BA

o

o o o o o o o

28 Cho ABC co A 110 . Tnh B va C e ABCco truc oi xng .

A. B = 50 va C 20 B. B = 45 va C 25 C. B = 40 va C 30 D. B = C 35

o o

o o oo

HD : Chon D . V : ABC co truc oi xng khi ABC can hoac eu

V A 110 90 ABC can tai A , khi o :

180 A 180 110B C 35

2 2

29 Trong cac hnh sau , hnh nao co nhieu truc oi xng nhat ?A. Hnh ch nhat B. Hnh vuong C. Hnh thoi D. Hnh thang can .S : Chon B. V : Hnh vuong co 4 truc oi xng .

30 Trong cac hnh sau , hnh nao co t truc oi xng nhat ?A. Hnh ch nhat B. Hnh vuong C. Hnh thoi D. Hnh thang can .S : Chon D. V : Hnh thang can co 1 truc oi xng .


14/35

- 14 -

31 Trong cac hnh sau , hnh nao co 3 truc oi xng ?A. Hnh thoi B. Hnh vuong C. eu D. vuong can .

S : Chon C. V : eu co 3 truc oi xng .

32 Trong cac hnh sau , hnh nao co nhieu hn 4 truc oi xng ?A. Hnh vuong B. Hnh thoi C. Hnh tron D. Hnh thang can .S : Chon C. V : Hnh tron co vo so truc oi xng .

33 Trong cac hnh sau , hnh nao khong co truc oi xng ?A. Hnh bnh ha nh B. eu C. can D. Hnh thoi .S : Chon A. V : Hnh bnh hanh khong co truc oi xng .

34 Cho hai hnh vuo

ng ABCD va AB C D co canh eu bang a va co nh A chung .Chng minh : Co the thc hien mot phep oi xng truc bien hnh vuong ABCD thanh AB C D .

HD : Ga s : BC B C = E .

AE

Ta co : AB = AB , B B 90 ,AE chung .EB = EB

ABE = AB F B Bbiet AB = AB

I

AE

A AE

EC = ECMat khac : C CAC = AC = a 2

BABNgoa i ra : AD = AD va D AE DAE 90

2

D D ABCD AB C D

I

I I

35 Goi H la trc tam ABC . CMR : Bon tam giac ABC , HBC , HAC , HAC cong tron ngoai tiep bang nhau .

1 21 1 1 2

BC BC

HD :

Ta co : A = C (cung chan cung BK )

A = C (goc co canh tng ng ) C = CCHK can K oi xng vi H qua BC .

Xet phep oi xng truc BC .

Ta co : K H ; B B ;I I

BC

BC

C C

Vay : ng tron ngoa i tiep KBC ng tron ngoa i tiep HBC

I

I

a

36 Cho ABC va ng thang a i qua nh A nhng khong i qua B,C .a) Tm anh ABC qua phep oi xng .

b) Goi G la trong tam ABC , Xac nh G la anh cua G qua phep oi xng a.

a

a

a

a

Giaia) V a la truc cua phep oi xng nen :

A a A (A) .

B,C a nen : B B ,C C sao cho a la trung trc cua BB ,CC

b) V G a nen : G G sao cho a la trung trc

I I

I cua GG .

37 Cho ng thang a va hai iem A,B nam cung pha oi vi a . Tm tren ngthang a iem M sao cho MA+MB ngan nhat .

Giai : Xet phep oi xng : A A .aM a th MA = MA . Ta c

I

o : MA + MB = MA + MB A Be MA + MB ngan nhat th chon M,A,B thang hang

Vay : M la giao iem cua a va A B .


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- 15 -

38 (SGK-P13)) Cho goc nhon xOy va M la mot iem ben trong goc o . Haytm iem A tren Ox va iem B tren Oy sao cho MBA co chu vi nho nhat .

GiaiGoi N = (M) va P = (M) . KhiOx Ox

o : AM=AN , BM=BPT o : CVi = MA+AB+MB = NA+AB+BP NP

( ng gap khuc ng thang )MinCVi = NP Khi A,B lan lt la giao iem cua NP vi Ox,Oy .

39 Cho ABC can tai A vi ng cao AH . Biet A va H co nh . Tm tap hpiem C trong moi trng hp sau :

a) B di ong tren ng thang .b) B di ong tren ng tro

n tam I, ban knh R .Giaia) V : C = (B) , ma B nen C vi = ( )AH AH

Vay : Tap hp cac iem C la ng thangb) Tng t : Tap hp cac iem C la ng tron tam J , ban knh R la anh cua

ng tron (I) qua .AH

Vn 4: PHP I XNG TMA.KIN THC C BN

1 N : Phep oi xng tam I la mot phep di hnh bien moi iem M thanh iem M oi xng vi M qua I.Phep oi xng qua mot iem con goi la phep oi tam .iem I goi la tam cua cua phep oi xng hay n gian la tam oi xng .

K hieu : (M) M IM IM .I

Neu M I th M INeu M I th M (M) I la trung trc cua MM .IN :iem I la tam oi xng cua hnh H (H) H.I

Chu y : Mot hnh co the khong co tam oi xng .

I2 Bieu thc toa o : Cho I(x ;y ) va phep oi xng tam I : M(x;y) M (M) (x; y ) tho o Ix = 2x xoy 2y yo

3 Tnh chat :1. Phep oi xng tam bao toan khoang cach gi

I

a hai iem bat k .2. Bien mot tia thanh tia .3. Bao toan tnh thang hang va th t cua cac iem tng ng .

4. Bien mot oan thang thanh oan thang bang no .5. Bien mot ng thang thanh mot ng thang song song hoac trung vi ng thang a cho .6. Bien mot goc thanh goc co

so o bang no .

7. Bien tam giac thanh tam giac bang no . ( Trc tam trc tam , trong tam trong tam )

8. ng tron thanh ng tron bang no . ( Tam bien thanh tam : I I , R = R )I

B . BI TP

1 Tm anh cua cac iem sau qua phep oi xng tam I :1) A( 2;3) , I(1;2) A (4;1)

2) B(3;1) , I( 1;2) B ( 5;3)3) C(2;4) , I(3;1) C (4; 2)


16/35

- 16 -

Giai :

x 1 3 x 4a) Ga s : A (A) IA IA (x 1;y 2) ( 3;1) A (4;1)I y 2 1 y 1

Cach : Dung bieu thc toa o

2 Tm anh cua cac ng thang sau qua phep oi xng tam I :1) ( ) : x 2y 5 0,I(2; 1) ( ) : x 2y 5 0

2) ( )

: x 2y 3 0,I(1;0) ( ) : x 2y 1 0

3) ( ) :3x 2y 1 0,I(2; 3) ( ) :3x 2y 1 0

GiaiPP : Co 3 cachCach 1: Dung bieu thc toa oCach 2 : Xac nh dang // , roi dung cong thc tnh khoang cach d( ; ) .Cach 3: Lay bat ky A,B , roi tm anh A ,B

I

A B

x 4 x x 4 x1) Cach 1: Ta co : M(x;y) M

y 2 y y 2 yI

I

V M(x;y) x 2y 5 0 (4 x ) 2( 2 y ) 5 0 x 2y 5 0

M (x ;y ) : x 2y 5 0

Vay : ( ) ( ) : x 2y 5 0Cach 2 : Goi = ( ) song songI

I

: x + 2y + m = 0 (m 5) .

|5| | m | m 5 (loai)Theo e : d(I; ) = d(I; ) 5 | m |

m 52 2 2 21 2 1 2

( ) : x 2y 5 0

Cach 3 : Lay : A( 5;0),B( 1; 2) A (9; 2),B (5;0) A B : x 2y 5 0

3 Tm anh cua cac ng tron sau qua phep oi xng tam I :2 2 2 21) (C) : x (y 2) 1,E(2;1) (C ) : (x 4) y 1

22) (C): x

2 2 2y 4x 2y 0,F(1;0) (C ) : x y 8x 2y 12 0

/ nghia hay bieu thc toa o23) (P) : y = 2x x 3 , tam O(0;0) .

E

2(P ) : y = 2x x 3

HD:1) Co 2 cach giai :Cach 1: Dung bieu thc toa o .

Cach 2 : Tm tam I I ,R R (a cho) .2) Tng t .

4 Cho hai iem A va B .Cho biet phep bien oi M than

I

h M sao cho AMBM la mot hnh bnh hanh .

HD :

MA BMNeu AMBM la hnh bnh hanh

MB AM

V : MM MA AM MA MB (1)

Goi I la trung iem cua AB . Ta co : IA IB

T (1) MM MI

IA MI IB MM 2MI

MI IM M (M) .I

5 Cho ba ng tron bang nhau (I ;R),(I ;R),(I ;R) tng oi tiep1 2 3xuc nhau tai A,B,C . Ga s M la mot iem tren

ICA B 1

(I ;R) , ngoai ra :1

M N ; N P ; P Q . CMR : M Q .I I I I


17/35

- 17 -

A A A

HD :

Do (I ;R) tiep xuc vi (I ;R) tai A , nen :1 2

M N ;I I MI NI MI NI (1)1 2 1 2 1 2I I I

B B B

C C C

Do (I ;R) tiep xuc vi (I ;R) tai B , nen :2 3

N P ;I I NI PI NI PI (2)2 3 2 3 2 3Do (I ;R) tiep xuc vi (I ;R) tai C , nen :

3 1 P Q ;I I PI3 1 3

I I I

I I I

1

QI PI QI (3)1 3 1

T (1),(2),(3) suy ra : MI QI M (Q) .1 1 I

5 Cho ABC la tam giac vuong tai A . Ke ng cao AH . Ve pha

ngoai tam giac hai hnh vuong ABDE va ACFG .

a) Chng minh tap hp 6 iem B,C,F,G,E,D co mot truc oi xng .

b) Goi K la trung iem cua EG . Chng minh K tren ng than

g AH .

c) Goi P = DE FG . Chng minh P tren ng thang AH .

d) Chng minh : CD BP, BF CP .e) Chng minh : AH,CD,BF ong qui .

DF DF DF DF

DF

HD :

a) Do : BAD 45 va CAF 45 nen ba iem D,A,F thang hang .

Ta co : A A ; D D ; F F ; C G ;

B E (Tnh chat hnh vuong ).Vay : Tap

l l l l

l

hp 6 iem B,C,F,G,E,D co truc oi xng chnh la ng thang DAF .

b) Qua phep oi xng truc DAF ta co : ABC = AEG nen BAC AEG.

Nhng : BCA AGE ( 2 oi xng = ) AGE A (do KAG can tai K) . Suy ra : A A K,A,H thang hang K tren AH .2 1 2c) T giac AFPG la mot hnh ch nhat nen : A,K,P thang hang . (Hn na K la trung iem cua AP )

Vay : P tren PH .

d) Do EDC = DBP nen DC = BP .DC = BP

Ta co : DB = AB BDC ABP CD BP BCD APB nhng hai goc nay co capBC = AP

canh : BC AP cap canh con lai : DC BP.Ly lua

n tng t , ta co : BF CP.

e) Ta co : BCP . Cac ng thang AH, CD va BF chnh la ba ng cao cua BCP nen ong qui .

2AB

6 Cho hai iem A va B va goi va lan lt la hai phep oi xng tam A va B .A Ba) CMR : T .B A

b) Xac nh .A BHD : a) Goi M la mot iem bat ky , ta co :

M

A

B

M : MA AM

M M : MB BM . Ngha la : M = (M), M (1)B A

I

I


18/35

- 18 -

B A Ta chng minh : M M :

Biet : MM MM M M

Ma : MM 2MA va M M 2M B

Vay : MM 2MA 2M B 2MA 2M A 2AB

V:MA

I

2AB

AM nen MA M A 0 . Suy ra : MM 2AB M T (M), M (2)

2AB

2BA

T (1) va (2) , suy ra : T .B A

b) Chng minh tng t : T .A B

7 Chng minh rang neu hnh (H) co hai truc oi xng vuong goc vi nhau th(H) co tam oi xng .

HD : Dung hnh thoiGa s hnh (H) co hai truc oi xng vuong goc vi nhau

.

Lay iem M ba t ky thuoc (H) va M (M) , M (M ) . Khi o , theo1 a 2 b 1

nh ngha M ,M (H) .1 2

Goi O = a b , ta co : OM = OM va MOM 2AOM1 1 1OM = OM va M OM 2M OB1 2 1 2 1

Suy ra : OM = OM va MOM M OM 2(AOM +M OB)2 1 1 2 1 1

hay MOM 2 90 1801Vay : O la

trung iem cua M va M .2Do o : M (M), M (H), M (H) O la tam oi xng cua (H) .2 O 2

N

8 Cho ABC co AM va CN la cac trung tuyen . CMR : Neu BAM BCN = 30 th ABC eu .HD :

T giac ACMN co NAM NCM 30 nen noi tiep tron tam O, bknh R=AC va MON 2NAM 60 .

Xet : A I

N

M M

B (O) (O ) th B (O ) v A (O) .1 1

C B (O) (O ) th B (O ) v C (O) .2 2

OO OO 2R1 2Khi o , ta co : OO O la tam giac eu .1 2MON 60

V O B O B R R 2R O O nen B la trun1 2 1 2

I

I I

g iem O O .1 2Suy ra : ABC OO O (V cung ong dang vi BMN) .1 2V OO O la tam giac eu nen ABC la tam giac eu .1 2

Vn 5: PHP QUAYA. KIN THC C BN

1 N : Trong mat phang cho mot iem O co nh va goc lng giac . Phep bien hnh bien moi iemM thanh iem M sao cho OM = OM va (OM;OM ) = c goi la phep quay tam O vi

Phep quay hoan toan xac nh khi biet tam va goc quay

K hieu : Q .O

goc quay .


19/35

- 19 -

Chu y : Chieu dng cua phep quay chieu dng cua ng tron lng giac .2kQ phep ong nhat , k(2k+1)Q phep oi xng tam I , k

2 Tnh chat :L : Phep quay la mot phep di hnh .HQ :

1.Phep quay bien ba iem thang hang thanh ba iem thang hang va bao toan th t cua cac iem tngng .

2. ng thang thanh ng th

ang .3. Tia thanh tia .4. oan thang thanh oan thang bang no .

(O ; )

Q Q5. Tam giac thanh tam giac bang no . (Trc tam trc tam , trong tam trong tam )

Q6. ng tron thanh ng tron bang no . ( Tam bien thanh tam : I I , R

I I

I = R )

7. Goc thanh goc bang no .

B. BI TP

(O ; )

/1 Trong mat phang Oxy cho iem M(x;y) . Tm M = Q (M) .(O ; )HD :

x = rcosGoi M(x;y) . at : OM = r , goc lng giac (Ox;OM) = th My = rsin

Q/ /V : M M . Goi M (x ;y ) th oI

/ / dai OM = r va (Ox;OM ) = + .Ta co :

x = rcos( + ) = acos .cos asin .sin x cos ysin .

y = rsin( + ) = asin .cos acos .sin xsin y cos .

x = x cos ysin/Vay : My = xsin y cos

(O ; )

(I ; )

o o

(I ; )

o o

ac biet :Q x = x cos ysin/ /M M

y = xsin y cos

Q x x = (x x )cos (y y )sin/ o o oM My y = (x x )sin (y y )cosI(x ;y ) o o o

Q

M I(x ;y )

I

I

I

x x = (x x )cos (y y )sin/ / o o oM y y = (x x )sin (y y )coso o o

(O;45 )

2 Trong mpOxy cho phep quay Q . Tm anh cua :(O;45 )

2 2a) iem M(2;2) b) ng tron (C) : (x 1) + y = 4Q

/ / /Giai . Goi : M(x;y) M (x ;y ) . Ta co : OM = 2 2, (Ox; OM)I

=

x = rcos( +45 ) r cos .cos45 rsin .sin45 x.cos45 y.sin45/Th My = rsin( +45 ) rsin .cos45 r cos .sin45 y.cos45 x.sin45

2 2x = x y

/ 2 2M2 2

y = x y2 2


20/35

- 20 -

(O;45 )

(O ; 45 )

(O ; 45 )

Q/

a) A(2;2) A (0 ;2 2)

Q/Tam I(1;0) Tam I ?b) V (C) : (C ) :

Bk : R = 2 Bk : R = R = 2

Q2 2 2 2/ 2 2I(1;0) I ( ; ) . Vay : (C ) : (x ) + (y ) =

2 2 2 2

I

I 4

1 3x = x y

2 23 Trong mpOxy cho phep bien hnh f : . Hoi f la phep g ?3 1

y = x y2 2

Giai

x = x cos ysin3 3Ta co f : M(x;y) M (x ;y ) vi f la phep quay Q

(O; )y = xsin y cos33 3

I

4 Trong mpOxy cho ng thang ( ) : 2x y+1= 0 . Tm anh cua ng thang qua :a) Phep oi xng tam I(1; 2). b) Phep quay Q .

(O;90 )Giai

a) Ta co : M (x ;y ) = (M) th bieu thcI

x 2 x x 2 xtoa o My 4 y y 4 y

V M(x;y) ( ) : 2x y+1= 0 2(2 x ) ( 4 y ) 1 0 2x y 9 0M (x ;y ) ( ) : 2x y 9 0

I

(O;90 )

Vay : ( ) ( ) : 2x y 9 0

Q

b) Cach 1 : Goi M(x;y) M (x ;y ) . at (Ox ; OM) = , OM = r ,Ta co (Ox ; OM ) = + 90 ,OM r .

x = rcosKhi o : M

y

I

I

(O;90 )

(

Qx r cos( 90 ) r sin y x y

M= rsin y xy rsin( 90 ) rcos x

V M(x;y) ( ) : 2(y ) ( x ) + 1 = 0 x 2y + 1 = 0 M (x ;y ) ( ) : x 2y 1 0Q

Vay : ( )

I

IO;90 ) ( ) : x 2y 1 0

(O;90 )

(O;90 )

(O;90 )

Q

Cach 2 : Lay : M(0;1) ( ) M ( 1; 0) ( )Q1 1

N( ;0) ( ) N (0; ) ( )2 2

Q

( ) ( ) M N : x 2y 1 0

I

I

I

(O;90 )

(O;90 )

Q1

Cach 3 : V ( ) ( ) ( ) ( ) ma he so goc : k 2 k2

Q

M(0;1) ( ) M (1;0) ( )

Qua M (1;0)

( ) : ( )1hsg ; k =

2

I

I

: x 2y 1 0


21/35

- 21 -

5 Trong mat phang toa o Oxy cho A(3;4) . Hay tm toa o iem A la anhocua A qua phep quay tam O goc 90 .

HD :

Goi B(3;0),C(0;4) lan lt la hnh chieu cua A len cac truc Ox,

Oy . Phepoquay tam O goc 90 bien hnh ch nhat OABC thanh hnh ch nhat OC A B .

Khi o : C (0;3),B ( 4;0). Suy ra : A ( 4;3).

6 Trong mat phang toa o Oxy . Tm phep quay Q bien iem A( 1;5)thanh iem B(5;1) .

OA OB 26HD : Ta co : OA ( 1;5) va OB (5;1)OA.OB 0 OA OB

B = Q(

(A) .O ; 90 )

7 Trong ma t phang toa o Oxy , cho iem M(4;1) . Tm N = Q (M) .(O ; 90 )

HD :

V N = Q (M) (OM;ON) 90 OM.ON = 0 4x+y = 0 y= 4x (1)(O ; 90 )

2 2Do : OM ON x y 16 1 17 (2) .

Giai (1) va

(2) , ta co : N(1; 4) hay N( 1;4) .

Th lai : ieu kien (OM;ON) 90 ta thay N( 1;4) thoa man .

8 a)Trong mat phang toa o Oxy , cho iem A(0;3) . Tm B = Q (A) .(O ; 45 )

HD : Phep quay Q bien iem A Oy thanh iem B t : y x,ta co :(O ; 45 )

x y 0 2 2B B . Ma OB = x y 3 xB BOA OB 3

o

3 3 3B( ; ).B

2 2 2

4 3 3 3 4 3b) Cho A(4;3) . Tm B = Q (A) B ( ; )(O;60 ) 2 2

2 29 Cho ng tron (C) : (x 3) (y 2) 4 . Tm (C ) = Q (C) .(O ; 90 )

2 2HD : Tm anh cua tam I : Q (I) I ( 2;3) (C ) : (x 2) (y 3) 4 .(O ; 90 )

2 210 Cho ng tron (C) : (x 2) (y 2 3) 5 . Tm (C ) =

Q (C) .(O ; 60 )

2 2HD : Tm anh cua tam I : Q (I) I ( 2;2 3) (C ) : (x 2) (y 2 3) 5 .(O ; 60 )

2 211 Cho ng tron (C) : (x 2) (y 2) 3 . Tm (C ) = Q (C) .(O ; 45 )

2 2HD : Tm anh cua tam I : Q (I) I (1 2;1 2) (C ) : (x 1 2) (y 1 2) 3 .(O ; 45 )

12 [CB-P19] Trong mat phang toa o Oxy , cho iem A(2;0) va ng thang (d) : x + y 2 = 0.Tm anh cua A va (d) qua phep quay Q .

(O ; 90 )

HD :

Ta co : A(2;0) Ox . Goi B = Q ((O ; 90 )

A) th B Oy va OA = OB .

V toa o A,B thoa man pt (d) : x + y 2 = 0 nen A,B (d) .Do B = Q (A) va tng t Q (A) = C( 2;0)(O ; 90 ) (O ; 90 )

x y x ynen Q (d) = BC (BC) : 1

(O ; 90 ) x y 2 2C C

1 x y 2 0


22/35

- 22 -

13 Cho (d) : x 3y 1 = 0 . Tm = Q (d) . ( ) : 3x y 1 0(O ; 90 )

14 Cho (d) : 2x y 2 = 0 . Tm = Q (d) .(O ; 60 )

1 3anhHD : d Ox = A(1;0) , d Oy = B(0;2) A ( ; ),B ( 3;1)

2 2

( ) : ( 3 2 )x (2 3 1)y 4 0

15 Cho tam giac eu ABC co tam O va phep quay Q .(O;120 )a) Xac nh anh cua cac nh A,B,C .b) Tm anh cua ABC qua phep quay Q

(O;120 )

Giai

a) V OA = OB = OC va AOC BOC COA 120 nen Q : A B,B C,C A(O;120 )

b) Q : ABC ABC(O;120 )

I I I

16 [CB-P19] Cho hnh vuong ABCD tam O .a) Tm anh cua iem C qua phep quay Q .

(A ; 90 )b) Tm anh cua ng thang BC qua phep quay Q

(O ; 90 )

HD : a) Goi E = Q (C) th AE=AC va(A ; 90 )

CAE 90 nen AEC

vuong can nh A , co ng cao AD . Do o : D la trung iem cua EC .b) Ta co : Q (B) C va Q (B) C Q (BC) CD.

(O ; 90 ) (O ; 90 ) (A ; 90 )

17 Cho hnh vuong ABCD tam O . M la trung iem cua AB , N la trung iemcua OA . Tm anh cua AMN qua phep quay Q .

(O;90 )

HD : Q (A) D , Q (M) M la trung iem cua A(O;90 ) (O;90 )

D .

Q (N) N la trung iem cua OD . Do o : Q ( AMN) DM N(O;90 ) (O;90 )

18 [ CB-1.15 ] Cho hnh luc giac eu ABCDEF , O la tam ng tron ngoai tiep cua no . Tm anh cua

OAB qua phep di hnh co c bang cach thc hien lien tiep phep quay tam O

OE

OE (O;60 )

(O;60 ) (O;60 ) (O;60 )

OE OE OE

, goc 60 va pheptnh tien T .

HD :

Goi F = T Q . Xet :

Q (O) O,Q (A) B,Q (B) C .

T (O) E,T (B) O,T (C) D

Vay : F(O) = E , F(A) = O , F(B) = D F( OAB) = EOD

19 Cho hnh luc giac eu ABCDEF theo chieu dng , O la tam ng tron ngoai tiep cua no . I latrung iem cua AB .

a) Tm anh cua AIF qua phep quay Q .(O ; 120 )

b) Tm a

nh cua AOF qua phep quay Q .(E ; 60 )

HD :

a) Q bien F,A,B lan l t thanh B,C,D , trung iem I(O ; 120 )thanh trung iem J cua CD nen Q ( AIF) CJB .

(O ; 120 )

b) Q bien(E ; 60 )

A,O,F lan lt thanh C,D,O .


23/35

- 23 -

15 Cho ba iem A,B,C theo th t tren thang hang . Ve cung mot pha dng hai tam giac eu ABE vaBCF . Goi M va N tng ng la hai trung iem cua AF va CE . Chng minh rang : BMN la tam giac eu .HD :Xet phep quay Q .Ta co : Q (A) E , Q (F) C

(B; 60 ) (B; 60 ) (B; 60 )Q (AF) EC .

(B; 60 )Do M la trung iem cua AF , N la trung iem cua EC , nen :

Q (M) N BM(B; 60 )

= BN va MBN 60 BMN la tam giac eu .

21 [ CB-1.17 ] Cho na ng tron tam O ng knh BC . iem A chay tren na ng tron o .Dng ve pha ngoai cua ABC hnh vuong ABEF . Chng minh rang : E chay

tren na

ng co nh .HD : Goi E = Q (A) . Khi A chay tren na ng tron (O) ,

(B;90 )

E se chay tren na ng tron (O ) = Q [(O)] .(B;90 )

22 Cho ng (O;R) va ng thang khong cat ng tron . Haydng anh cua ( ) qua phep quay Q .

(O ; 30 )

Giai

T O ha ng vuong goc OH vi . Dng iem H sao cho(OH

;OH ) = 30 va OH = OH . Dng ng tron qua 3 iem O,H,H ;

ng tron nay cat tai iem L . Khi o LH la ng thang phai dng .

23 Cho ng thang d va iem O co nh khong thuoc d , M la iemdi ong tren d . Hay tm tap hp cac iem N sao cho OMN eu .

Giai : OMN eu OM ON va NOM 60 . V vay khi M cha

y tren d th :N chay tren d la anh cua d qua phep quay Q .

(O;60 )

N chay tren d la anh cua d qua phep quay Q(O; 60 )

24 Cho hai ng tron (O) va (O ) bang nhau va cat nhau A va B .T iem I co nh ke cat tuyen di ong IMN vi (O) , MB va NB cat(O ) tai M va N . Chng minh ng thang

M N luon luon i qua motiem co nh.

GiaiXet phep quay tam A , goc quay (AO; AO ) = bien (O) thanh (O ) .V MM va NN qua B nen (AO;AO ) = (AM;AM ) = (AN;AN ) .

Qua phep quay Q : MI

(A; )

M , N N va do oQ

MN M N

ng thang MN qua iem co nh I nen ng thang M N quaiem co nh I la anh cua I qua Q(A; )

I

I

25 Cho hai hnh vuong ABCD va BEFGa) Tm anh cua ABG trong phep quay Q .

(B; 90 )

b) Goi M,N lan lt la trung iem cua AG va CE .Chng minh BMN vuong can .

GiaiBA BC

a) V(BA;

BG BEva

BC) 90 (BG;BE) 90

Q : A C,G E Q : ABG CBE(B; 90 ) (B; 90 )

b) Q : AG CE Q : M N BM BN va (BM;BN) = 90(B; 90 ) (B; 90 )

BMN vuong can tai B .

I I

I


24/35

- 24 -

26 Cho ABC . Qua iem A dng hai tam giac vuong can ABE va ACF . Goi M la trung iem cua BCva gia s AM FE = H . Chng minh : AH la ng cao cua AEF .

HD :

Xet phep quay Q : Keo da i FA mot oan AD = AF .(A;90 )

V AF = AC AC = AD nen suy ra : Q bien B , C lan lt thanh E , D(A;90 )

/ngh

nen goi trung iem K cua DE th K= Q (M)(A;90 )

a

MA AK (1) .

Trong DEF , v AK la ng trung bnh nen AK // FE (2)T (1),(2) suy ra : AM FE AH la ng cao cua AEF .

27 Cho hnh vuong ABCD co canh bang 2 va co cac nh ve theo chieudng . Cac ng cheo cat nhau tai I. Tren canh BC lay BJ = 1 . Xac nh

phep bien oi AI thanh BJ .

HD

AB 2: Ta co : AI= 1 AI BJ . Lai co : (AI,BJ) 45 .

2 2

BJ = Q (AI) . Tam O = ttrc cua AB cung cha goc 45 i(O;45 )

qua A,B BJ = Q (AI)(O;45 )

28 [CB-1.18] Cho ABC . Dng ve pha ngoai cua tam giac cac hnh vuong BCIJ,ACMN,ABEFva goi O,P,Q lan lt la tam oi xng cua chung .

a) Goi D la trung iem cua AB . Ch

ng minh rang : DOP vuong can tai D .b) Chng minh rang : AO PQ va AO = PQ .

HD :

a) V : AI = Q (MB) MB = AI va MB AI .(C;90 )

Mat khac : DP1

BM , DO2

AI

DP = va DO DOP vuong can tai D .

(D;90 ) (D;90 )

b) T cau a) suy ra :Q Q

O P,A Q OA va PQ.I I


25/35

- 25 -

29 Cho ABC co cac nh k hieu theo hng am . Dngve pha ngoai tam giac o cac hnh vuong ABDE va BCKF .Goi P la trung iem cua AC , H la iem oi xng cua D qua B ,M la tr

ung iem cua oan FH .

a) Xac nh anh ua hai vect BA va BP trong phep quay Q .(B;90 )

b) Chng minh rang : DF BP va DF = 2BP .

HD :BA = BH (cung bang BD)

a) Ta co :(BA;BH) = 90

90 90H Q (A) BH Q (BA)B B

90 90 90V : Q (A) H,Q (C) F Q (AC) HF .B B B90 90Ma : F la trung iem cua AC , Q (F) M la trung iem cua HF . Do o : Q (BP) BMB B

.

90b) V : Q (BP) BM BP BM,BP BM .B

1 1Ma : BM = DF va BM // DF (ng trung bnh cua HDF ). Do o : BP = DF , DF BP .

2 2

30 Cho t giac loi ABCD . Ve pha ngoai t giac dng cac tam giac eu ABM , CDP . Ve pha trongt giac, dng hai tam giac eu BCN va ADK . Chng minh : MNPK la hnh bnh hanh .

H

(B;90 )

(D;90 )

60D : Xet phep quay Q : M A , N CBQ

MN AC MN AC (1)

60

Xet phep quay Q : P C , K ADQ

PK CA PK CA (2)

T (1) , (2) suy ra : MN = PK .L luan , t

I I

I

I I

I

ng t : MK = PN MKNP la hnh bnh hanh .

(B;60 ) (B;60 )

31 Cho ABC . Ve pha ngoai tam giac , dng ba tam giac euBCA ,ACB ,ABC . Chng minh rang : AA ,BB ,CC ong quy .1 1 1 1 1 1

HD :

Q Q

Ga s AA CC I . Xet : A C,A C1 1 1 1

A A1

I I

I

(B;60 )Q

CC (A A;CC ) 60 AJC 60 (1)1 1 1 1

Lay tren CC iem E sao cho : IE = IA . V EIA 60 EIA eu .1

(A;60 ) (A;60 ) (A;60 )Q Q Q

Xet : B C ,I E , B C1 1V : C ,B,C thang hang nen B,I,B thang hang1 1

AA ,BB ,CC ong quy .1 1 1

I I I


26/35

- 26 -

32 Chng minh rang cac oan thang noi tam cac hnh vuong dngtren cac canh cua mot hnh bnh hanh ve pha ngoai , hp thanhmot hnh vuong .

HD : Goi I , I , I , I la tam cua1 2 3 4

(I;90 )

hnh vuong canh AB,BC,CD,DA .

Dung phep quay Q(I;90 ) : B C . V I BA I CD1 3

CI BI va DCI ABI 45 . Ma DC // AB CI BI3 1 3 1 3 1Q

Vay : I I I I I I va I I I I .3 1 2 1 2 3 2 1 2 3Ly luan tng t

I

I

, ta co : I I I I la mot hnh vuong .1 2 3 4

Vn 6: HAI HNH BNG NHAUA. KIN THC C BN

1 L : Neu ABC va A B C la hai tam giac bang nhau th co phep di hnh bien ABC thanh A B C .

2 Tnh chat :

1. Neu thc hien lien tiep hai phep di hnh th c mot phep di hnh .2. Hai hnh goi la bang nhau neu co phep di hnh bien hnh nay thanh hnh kia .

B. BI TP

1 Cho hnh ch nhat ABCD . Goi E,F,H,I theo th t la trung iem cua cac canhAB,CD,BC,EF. Hay tm mot phep di hnh bien AEI thanh FCH .

HD :

Thc hien lien tiep phep tnh tie

n theo AE va phep oi xng qua ng thang IH

T : A E,E B,I H T ( AEI) EBHAE AE

I I I

: E F,B C,H H ( EBH) FCHIH IH : T ( AEI) FCHIH AEDo o : T ( AEI) FCH AEI FCHIH AE

I I I

2 Cho hnh ch nhat ABCD . Goi O la tam oi xng cua no ; E,F,G,H,I,J theo th t la trung iem cuacac canh AB,BC,CD,DA,AH,OG . Chng minh rang : Hai hnh thang AJOE va GJFC bang nhau .

HD :

Phep tnh tien theo AO bien A,I,O,E lan lt thanh O,J,C,F . Phep oixng qua truc cua OG bien O,J,C,F lan lt thanh G,J,F,C.

T o suy ra phep di hnh co c bang cach thc hien lien tiep haiphep bien hnh tren se bien hnh thang AJOE thanh hnh thang GJFC .Do o hai hnh thang ay bang nhau .

3 [CB-1.20] Trong mpOxy , cho u = (3;1) va ng thang (d) : 2x y = 0 . Tm anh cua (d) qua phepdi hnh co c bang cach thc hien lien tiep phep quay Q va phep tnh tien

(O;90 )T .u


27/35

- 27 -

(O;90 ) uQ

THD : PP : d d d

Goi d Q (d) . V tam O d nen Q (O) O d .(O;90 ) (O;90 )

Mat khac : d d d : x 2y C 0 (C 0) ma d qua O nen C = 0 d : x + 2y = 0

Cach khac : Chon

I I

(O;90 )Q

M(1;2) d M d .

x OMcos( 90 ) x OMcos cos90 OMsin sin90 x xcos90 ysin90Ta co : My OMsin( 90 ) y OMsin cos90 OMcos sin90 y y cos90 xsin90

I

x 1cos90 2sin90 x 2M ( 2;1)

y 1y 2cos90 1sin90

Goi d T (d ) d // d d : x 2y C 0 .ux x 3 x 3

Goi O T (O) OO = u O (3;1) .u y y 1 y 1V d O 3 2 C 0

C 5 d : x 2y 5 0

Vay :T Q (d) (d ) : x 2y 5 0u (O;90 )

2 24 Tm anh cua ng tron (C) : x y 2x 4y 4 0 co c bang cach thc hien lien tiep pheptnh tien theo u = (3; 1) va phep .Oy

2 2 S : (C ) : (x + 4) (y 3) 9

2 25 Tm anh cua ng tron (C) : x y 6x 2y 6 0 co c bang cach thc hien lien tiep phepquay Q va phep .Ox(O;90 )

HD : (C) co tam I(3;1) , bk : R = 2 . Khi o :

(C) : I(3;1)

(O;90 ) OxQ

, R = 2 (C ) : I ( 1;3) , R = 2 (C ) : I ( 1; 3) , R = 22 2(C ) :(x + 1) (y 3) 4

I I

6 [CB-P23] Trong mpOxy cho cac iem A( 3;2),B( 4;5) va C( 1;3).a) Chng minh rang : Cac iem A (2;3),B (5;4) va C (3;1) theo th t la anh cua A,B va C qua Q .

(O; 90 )

b) Goi A B C l1 1 1 a anh cua ABC qua phep di hnh co c bang cach thc hien lien tiep phep

(O;

Q va phep oi xng . Tm toa o cac nh cua A B C .Ox 1 1 1(O; 90 )HD :

a) Goi M,N lan lt la hnh chieu cua A tren Ox,Oy thM( 3;0),N(0;2).Q

Khi o : Hnh ch nhat OMAN I

90 )

(O; 90 )

hcnhat OM A Nvi M (0;3),N (2;0).

Do o : A (2;3) = Q (A) .(O; 90 )

Tt : B (5;4) = Q (B),C (3;1) = Q (C) .(O; 90 ) (O; 90 )

Q

Cach khac : Ga s A A AOA vuong cI

an tai O .

ieu o ung v : OA = OA = 13, OA.OA 0 . Lam tng t cho B,C ta co ieu can chng minh .

b) Phep quay : Q ( ABC) A B C , ( A B C ) A B COx 1 1 1(O; 90 )


28/35

- 28 -

1

1

x x 2A AKhi o : A (2; 3).Tt : B (5; 4),C (3; 1).1 1 1y y 3A A

27 Trong mpOxy , cho hai parabol : (P ) : y 2x ,12(P ) : y 2x 4x 1. Khang nh nao sau ay sai ?2

2 2A) y 2x 4x 1 y 2(x 1) 3

B)

Tnh tien sang trai 1 n v roi xuong di 3 n v ta c (P ).2C) (P ) va (P ) bang nhau .1 2D) Phep tnh tien theo u = (1; 3) bien (P ) thanh (P ) .1 2

S : B)

8 Trong mpOxy ,

(O;90 )

cho 4 iem A(2;0),B(4;4),C(0;2) va D( 4;4) .Khang nh nao sau ay sai ?A) Cac OAC, OBD la cac tam giac vuong can .

Q

B) Phep quay : OAB OCD .

C) OAB va OCD la hai h

I

nh bang nhau .D) Ton tai mot phep tnh tien bien A thanh B va C thanh D .

S : D)

9 Trong mpOxy cho ABC vi A( 3;0),B(0;3),C(2;4) . Phep bien hnh f bien A thanh A ( ;3) , Bthanh B (2;6),C thanh C (4;7) . Khang nh nao sau ay ung ?

3

A) f la phep quay Q . B) f la phep oi xng tam I( 1; ) .(O;90 ) 2

C) f la phep tnh tien theo vect u = (2;3) . D) f la phep oi xng truc .S : C)

Vn 7: PHP V T

1 N : Cho iem I co inh va mot so k 0 . Phep v t tam I t so k .kK hieu : V , la phep bien hnh bien moi iem M thanh iem M sao cho IM k IM.I

kI

k2 Bieu thc toa o : Cho I(x ;y ) va phep v t V .o o Ix = kx+ (1 k)xV k oM(x;y) M V (M) (x ;y ) thI y = ky+ (1 k)yo

I

3 Tnh chat :k k1. M V (M), N V (N) th M N = kMN , M N = |k|.MNI I

2. Bien ba iem thang hang thanh ba iem thang hang va bao toan th t cua cac iem tng ng .3. Bien mot ng thang thanh mot ng thang song song hoac trung vi ng thang a cho .

4. Bien mot tia thanh tia .5. Bien oan thang thanh oan thang ma o dai c nhan len |k| .6. Bien tam giac thanh tam giac ong dang vi no .7. ng tron co ban knh R tha nh ng tron co ban knh R = |k|.R .

8. Bien goc thanh goc bang no .

B . BI TP


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1 Tm anh cua cac iem sau qua phep v t tam I , t so k 0 :a) A(1;2) , I(3; 1) , k = 2 .

A ( 1;5)

b) B(2; 3),I( 1; 2),k 3 . B ( 10;1)

1c) C(8;3), I(2;1) , k = .

2

C (5;2)

2 1 1d) P( 3;2),Q(1;1),R(2; 4) , I O,k = 1/ 3 P (1; ),Q ( ; )

3 3 3

(I;2)

2 4,R ( ; )

3 3

V x 3 4HD : a) Goi : A(1;2) A (x ;y ) IA 2IA (x 3;y 1) 2( 2;3)y 1 6

x 1A ( 1;5) .

y 5

I

2 Cho ba iem A(0;3),B(2; 1),C( 1;5) . Ton tai hay khong ton tai mot phep v t tam A , t so k bienB thanh C ?

HD : Ga s ton tai mot phep v t tam A , t so k bien B tha

(A;k)

nh C .

V 11 k(2)Khi o : B C AC kAB k

2 k( 4) 2

Vay : Ton tai phep v t V : B C .1(A; )

2

3 Cho ba iem A( 1;2),B(3;1),C(4;3) . Ton tai hay khong ton ta

I

I

(A;k)

i mot phep v t tam A , t so k bienB thanh C ?

HD : Ga s ton tai mot phep v t tam A , t so k bien B thanh C .V

Khi o : B C AC kAB (1) .I

4 Cho OMN . Dng anh cua M,N qua phep v t tam O , t so k trong moi trng hp sau :

1 3a) k = 3 b) k = c) k =2 4

Giai3a) Phep v t V : M M , NO I I

N th ta co OM 3OM, ON 3ON

1/2b) Phep v t V : M H , N K th HK la ng trung bnh cua OMN .O33/4c) Phep v t V : M P , N Q th ta co OP OM,OQO 4

I I

I I 3

ON4


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5 Cho hnh bnh hanh ABCD (theo chieu kim ong ho) co tam O . Dng :a) Anh cua hnh bnh hanh ABCD qua phep v t tam O , t so k = 2 .

b) Anh cua hnh bnh hanh ABCD qua phep v t

1 tam O , t so k = .

2Giai

2a) Goi V : A A th OA 2OAO

B B th OB 2OBC C th OC 2OC

D D th O

I

I

I

I

D 2OC

2V : ABCDM A B C D .OTa ve : AB// A B ,BC//B C ,CD//C D ,DA // D A

11/2b) Goi V : A P th OP OAO 21

B Q th OQ OB2

I

I

I

1C R th OR OC

21

D S th OS OD2

1/2V : ABCDM PQRS .OTa ve : AB// PQ,BC// QR,CD// RS,DA // SP .

I

I

6 Cho ABC co AB = 4, AC = 6 , AD la phan giac trong cua A cua ABC (D BC) . Vi gia tr naocua k th phep v t tam D , t so k bien B thanh C .

HD :Theo tnh chat cua phan gi

( D; 3/2 )

ac trong cua A , ta co :VDB AB 4 2 3

DC DB B C .AC 6 3 2DC

Do DB va DC ngc hng .

I

7 Cho ABC vuong A va AB = 6, AC = 8 . Phep v t V bien B thanh B ,C thanh C .3(A; )

2Khang nh nao sau ay sai ?

9

A) BB C C la hnh thang . B) B C = 12 . C) S S . D) Chu vAB C ABC4

(A;3/2)

2

i ( ABC) = Chu vi( AB C ) .3HD :

VA) ung v B C BC .

3 3 2 2B) sai v : B C = BC AB AC 152 2

1 3 3.AB .AC .AB. .ACS 9AB C 2 2 2C) ung v : .

1S AB.AC 4ABC .AB.AC2

Chu vi AB C 3D) ung v :Chu vi ABC 2

8 Cho ABC co hai nh la B va C co nh , con nh A di ong tren ng tron (O) cho trc .Tm tap hp cac trong tam cua ABC .


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1

HD : Goi I la trung iem cua BC . Ta co I co nh . Neu G la trong tam cua ABC th IG IA .3

1/3Vay G la anh cua A qua phep v t V .I

Tap hp iem A la ng tron (O) nen tap hp G la ng tron (O ) , o chnh la anh cua ng tron1/3(O) qua phep v t V .I

9 Trong mpOxy , cho iem A( 1;2) va ng thang d

i qua A co he so goc bang 1 . Goi B la ng

thang di ong tren d . Goi C la iem sao cho t giac OABC la hnh bnh hanh .Tm phng trnh taphp :a) Cac tam oi xng I cua hnh bnh hanh .b) Cac trong tam G cac tam giac ABC .

HD :

a)

Qua A( 1;2)(AB): (AB) : y 2 1(x 1) y x 3

Hsg : k = 1

1Vay B chay tren d th I chay tren d // d va i qua trung iem M( ;1) cua oan OA .23

Vay d : x y = 0 .2

b) Ta

2 2 42/3 2/3co : OG OB G V (B) . Vay G chay tren t d // d va qua iem N( ; ) V (A).O O3 3 3d : x y 2 = 0 .

10 Tm anh cua cac ng thang d qua phep v t tam I , t so k :2

a) d : 3x y 5 = 0 ,V(O; ) d : 9x 3y 13

0 0

b) d : 2x y 4 = 0 ,V(O;3) d : 2x y 12 0c) d : 2x y 4 = 0 ,V(I; 2) vi I( 1;2)

d : 2x y 8 0

d) d : x 2y 4 = 0 ,V(I;2) v i I(2; 1) d : x 2y 8 0

11 Tm anh cua cac ng tron (C) qua phep v t tam I , t so k : (Co 2 cach giai )2 2a) (C) : (x 1) (y 2) = 5 ,V(O; 2) (C) : (x 2) 2 2(y 4) = 202 2 2 2b) (C) : (x 1) (y 1) = 4 ,V(O; 2) (C) : (x 2) (y 2) = 162 2c) (C) : (x 3) (y 1) = 5 ,V(I; 2) vi I(1;2)

2 2(C) : (x 3) (y 8) = 20

12 Tm phep v t bien d thanh d :

x ya) d : 1,d : 2x y 6 0,V(O; k)2 4

2k = .3

HD : d : 2x y 4 0 // d : 2x y 6 0 . Lay A(2;0) d,B(3;0) d .3

V : phep v t V(O;k) : A B OB kOA . V : OA= (2;0),OB (3;0) OB OA2

I

3 3V(O; ) V(O; )

2 2Vay : A B d dLu y : V O,A,B thang hang nen ta chon chung cung nam tren mot ng thang . e n gian ta chonchung cung nam tren Ox hoac Oy

I I

.


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2 2 2 2b) (C ) : (x 4) y 2 ; (C ) : (x 2) (y 3) 8 V(I; 2),I( 2;1)1 2HD :

(C ) co tam I ( 4; 0), R 2 , (C ) co tam I (2;3),R 2 21 1 1 2 2 2V(I;k)

Ga s :(C )1 I (C ) th :2

R2 R | k | R | k | 2 k 22 1

R1II kII th2 1

k = 2 . Goi I(x ;y ) th (2 x ;3 y ) 2( 4 x ; y ) I( 2;1)o o o o o ok = 2 . Goi I(x ;y ) th (2 x ;3 y ) 2( 4 x ; y ) I( 10; 3)o o o o o o

Vay co 2 phep v t bien (C ) (C ) la V(I; 2) vi I( 2;1) hoac V(I;2) v i I( 10; 3)1 2

2 2 2 213 Trong mpOxy , cho 2 ng tron (C ) : (x 1) (y 3) = 1 va (C ) : (x 4) (y 3) = 4 .1 2a) Xac nh toa o tam v t ngoai cua hai ng tron o .b) Viet phng trnh cac tiep tuyen c

hung ngoai cua hai ng tron o .

HD : (C ) co tam I (1;3) , bk : R 1 ; (C ) co tam I (4;3) , bk : R 2 .1 1 1 2 2 2

a) Goi I la tam v t ngoai cua (C ) va (C ) , ta co : II kII vi1 2 2 1 R 22k = 2 I( 2;3)R 11

b) Tiep tuyen chung ngoai cua hai ng tron la tiep tuyen t I en (C ).1Goi t i qua I va co he so goc k :y 3 = k(x+2) ky y 3 2k 0 .

1tiep xuc (C ) d(I ; ) R k1 1 1

2 2

: 2.x 4y 12 3 2 01

: 2.x 4y 12 3 2 02

14 Cho ng tron (O,R) ng knh AB . Mot ng tron (O ) tiep xuc vi (O,R) va oan AB tai

C, D , ng thang CD cat (O,R) tai I . Chng minh rang : AI BI .HD :

C la tam v

t cua 2 ng tron (O) va (O ) .D (O ), I (O) va ba iem C,D,I thang hang .Goi R la ban knh cua ng tron (O ) , khi o :

R

RV : O O ,I DC

OI // O D OI AB (V

I I

O D AB)

I la trung iem cua AB AI BI .

15 Cho hai ng tron (O,R) va (O , R ) tiep xuc trong tai A (R > R ) .ng knh qua A cat (O,R) tai B va cat (O , R ) tai C . Mot ng

thang di ong qua A cat (O, R) tai M va ca

t (O , R ) tai N . Tm quy tch cua I= BN CM .HD :

IC CNTa co : BM // CN . Hai BMI NCI . Do o :

IM BM


33/35

- 33 -

AC CNHai ACN ABM . Do o :

AB BMIC AC 2R R IC R

IM AB 2R R IM IC R RR

V(C;k )CI R R R RCI CM M : I

CM R R R RVay : Tap hp cac iem I la ng tron ( ) v t cua ng

I

R

tron (O,R) trong phep v t V(C ;k ) .R R

16 Cho ABC . Goi I , J . M theo th t la trung iem cua AB, AC va IJ . ng tron ngoai tiep tam Ocua AIJ , cat AO tai A . Goi M la chan ng vuong goc ha t A xuong BC

. Chng minh rang : A ,

M , M thang hang .

HD :

Goi M la trung iem BC .Ta co : AB 2AI va AC 2AJ1V(A;2)

T o : AIJ ABC . Khi o :

V : O A ,M M OM IJ A M BC .(A;2) 1 1

Nh the : M M A, M, M thang hang ( v A,M1

I I

,M thang hang )1

17 Cho ABC . Goi A ,B ,C tng ng la trung iem cua BC,CA,1 1 1AB. Ke A x,B y,C z lan lt song song vi cac ng phan giac trong1 1 1cua cac goc A,B,C cua ABC . Chng minh : A x,B y,C z1 1 1 ong quy.

HD :

1Xet phep v t tam G , t so . G la trong tam ABC ,

2

I la tam ng tron noi tiep ABC .Ta co : AJ A x , BI B y , CI C z ,1 1 1

GI 1I J ( ) A x,B y,C z ong quy ta1 1 1GJ 2

I I I

I i J .

18 Cho hai ng tron (O ,R ) va (O ,R ) ngoai nhau1 1 2 2R R . Mot ng tron (O) thay oi tiep xuc ngoai1 2

vi (O ) tai A va tiep xuc ngoai vi (O ) tai B . Chng1 2minh rang : ng thang AB luon luon i qua mot iemco nh .

HD :A la tam v t bien (O ) thanh (O) : AO va AO ngc hng .1 1B la tam v t bien (O) thanh (O ) : AO va AO ngc hng .2 1Keo dai AB cat (O ) tai C : AO va2 CO ngc hng .2Vay : AO va CO ngc hng . Nh vay AC hay cung la1 2AB phai i qua tam I a tam v t ngoai cua (O ) va (O ) .1 2


34/35

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19 Cho ABC . Ngi ta muon nh ba iem A ,B ,C lan lt tren cac canh BC,CA,AB sao cho A B Ceu va A B CA , B C AB va C A BC .

1. Goi E,F,K lan lt la chan cac ng cao

phat xuat t A,B,C .2/3 2/3 2/3at : C = V (A),A = V (E),B = V (F).B B B

22/3a) Nghiem lai rang : A = V (E) va B C CK .B 3b) Suy ra rang : A B C eu .

2. Chng minh rang trc

tam H cua ABC cung la trong tam cua A B C .HD :

a 3Trong ABC eu cac ng cao : AE = BF = CK = .(a la canh cua ABC)

2va E,F,K lan lt la trung iem cac canh .

1. a) V A = V

2 2 1 22/3 2/3(E) BA BE BC CA ( BC) CA CB . Vay : A = V (E) .B B3 3 2 32 2 1 22/3 2/3V C = V (A) BC BA BA AC BA AC BA AK B = V (C).B A3 3 3 3

2/3 2/3A AV V 2Vay : C B , K C B C CK .

3I I

B C //CK cung AB2

b) Ta co : B C CK 2 a 33 B C = CK =

3 3

2 2

Tng t : C A AE va A B BF .3 3

a 3

Vay : B C AB,C A BC,A B AC va B C =C A =A B = A B C eu .3

2. Trc tam H cua ABC cung la trong tam cua tam giac o , nen :2 2 2 2

BH BF. Ma : BC BA BH BC (BF BA) C H AF .3 3 3 3

Vay : C H // AF . Suy ra : C

H A B

Ly luan tng t : A H B C .

Vn 8: PHP NG DNGA. KIN THC C BN

1 N : Phep bien hnh F goi la phep ong dang t so k (k > 0) neu vi hai iem bat k M , N va anh M ,N la anh cua chung , ta co M N = k.MN .

2 L : Moi phep ong dang F t so k (k> 0) eu la hp thanh cua mot phep v t t so k va mot phepdi hnh D.

3 He qua : (Tnh chat ) Phep ong dang :1. Bien 3 iem thang hang thanh 3 iem thang hang (va bao toan th t ) .2. Bien ng thang thanh ng thang .3. Bien tia thanh tia .4. Bien oan thang thanh oan thang ma o dai c nhan len k ( k la t so on

g dang ) .5. Bien tam giac thanh tam giac ong dang vi no ( t so k).6. Bien ng tron co ban knh R thanh ng tron co ban knh R = k.R .7. Bien goc thanh goc bang no .4

Hai hnh ong dang :N : Hai hnh goi la ong dang vi nhau neu co phep ong bien hnh nay thanh hnh kia .

FH ong dang G F ong dang : H GI


35/35

B.BI TP

1 Cho iem Ma) Dng anh cua phep ong dang F la hp thanh cua phep oi xng truc va phep v t V tam O ,a

vi O a , t so k = 2 .b) Dng anh cua phep ong dang F la

2a O

hp thanh cua phep v t V tam O , t so k = 3 va phep quay

tam I vi goc quay = 90 .Giai

Va) Goi : M M M1 2M (a) th M M va M la trung iem OM1 2M (a) v

I I

a O M th :1 a la trung trc oan MM1

M la trung iem oan OM1 2M (a) va O M th :1

a la trung trc oan MM1M la trung iem oan OM1 2

b) Go

3 90O IV Qi M M M . Khi o :1 2OM 3OM , IM = IM va (IM ;IM) 901 1 1

I I

2 Cho ABC co ng cao AH . H tren oan BC . Biet AH = 4 , HB = 2 , HC = 8 . Phep ong dang Fbien HBA thanh HAC . F c hp thanh bi hai phep bien hnh nao di ay ?

A) P1

hep oi xng tam H va phep v t tam H t so k = .2

B) Phep tnh tien theo BA va phep v t tam H t so k = 2 .C) Phep v t tam H t so k = 2 va phep quay tam H , goc (H

B;HA) .

D) Phep v t tam H t so k = 2 va phep oi xng truc .HD :

2Phep V va Q(H; ) vi = (HB;HA) : B A, A CHVay : F la phep ong dang hp thanh bi V va Q bien HB

I I

A thanh HAC .

3 Cho hnh bnh hanh ABCD co tam O . Tren canh AB lay iem I sao cho IA 2IB 0 va goi G latrong tam cua ABD . F la phep ong dang bien AGI thanh COD . F c hp thanh

bi hai phepbien hnh nao sau ay ?A) Phep tnh tien theo GO va phep v t V(B; 1) .

1B) Phep oi xng tam G va phep v t V(B; ).

23

C) Phep v t V(A; ) va phep oi xng2 tam O .2

D) Phep v t V(A; ) va phep oi xng tam G .3

2/3

OA

HD :

3V G la trong tam ABD nen AO AG

23

Theo gia thie t , ta co : AB AJ .2

Phep oi xng tam O , bien A thanh C va B thanh D ( O la bat bien )

VA AI I

2/3 2/3O OA A V VC . G O O . I B D .I I I I

3V(A; )

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