PHÉP DỜI HÌNH VÀ PHÉP ĐỒNG DẠNG TRONG MẶT PHẲNG
Vần đề 1 : PHÉP BIẾN HÌNH- PHÉP DỜI HÌNH
B . BÀI TẬP
�
x�
=2x 1
1 Trong mpOxy cho phe�
p bie�
n h�
nh f: M(x;y) I��
� M�
=f(M) = �
. Ảnh của hai điểm
y�
=y +3
�
A(1;2), B( 1;2) là:
A. A �
(-1;5) , B�
( 7;6)
B. A�
(1;5) , B�
( 7;6)
C. A�
(1;5) , B�
( 7;-6) D. A�
(2;-5) , B�
(1;6)
�
x�
=2x y 1
2 Trong mpOxy cho phe�
p bie�
n h�
nh f : M(x;y) I��
� M�
=f(M) = �
.
�
y
� =x 2y +3
T�
m a�
nh cu�
a ca�
c�
ie�
m sau : a) A(2;1) b) B( 1;3) c) C( 2;4)
Gia�
i:
a) A �
=f(A) =(4;3)
�
b) B =f(B) =( 4; 4)
c) C�
=f(C) =( 7; 7)
3 Trong mpOxy cho phe�
p bie�
n h�
nh f : M(x;y) I��
� M�
=f(M) =(3x;y) . �a�
y co�
pha�
i la�
phe�
p d�
�
i
h�
nh hay kho�
ng ?
Gia�
i : La�
y hai �
ie�
m ba�
t k�M(x1;y1),N(x2;y2)
Khi �
o�
f : M(x1;y1) I��
� M�
=f(M) =(3x1; y1) .
f : N(x2;y2) I��
� N�
=f(N) =(3x2; y2)
Ta co�
: MN = (x2 x1)2 (y2 y1)2 , M ��
N = 9(x2 x1)2 (y2 y1)2
Ne�
u x1 �x2 th�M ��
N �MN . Va�
y : f kho�
ng pha�
i la�
phe�
p d�
�
i h�
nh .
(V�co�
1 so�
�
ie�
m f kho�
ng ba�
o toa�
n khoa�
ng ca�
ch) .
4 Trong mpOxy cho 2 phe�
p bie�
n h�
nh :
a) f : M(x;y) I��
� M�
=f(M) =(y ; x-2)
b) g : M(x;y) I��
� M�
=g(M) =( 2x ; y+1) .
Phe�
p bie�
n h�
nh na�
o tre�
n�
a�
y la�
phe�
p d�
�
i h�
nh ?
HD :
a) f la�
phe�
p d�
�
i h�
nh
b) g kho�
ng pha�
i la�
phe�
p d�
�
i h�
nh ( v�
x1 �x2 th�
M ��
N �MN )
5 Trong mpOxy cho 2 phe�
p bie�
n h�
nh :
a) f : M(x;y) I��
� M�
=f(M) =(y +1 ; x)
b) g : M(x;y) I��
� M�
=g(M) =( x ; 3y ) .
Phe�
p bie�
n h�
nh na�
o tre�
n�
a�
y la�
phe�
p d�
�
i h�
nh ?
Gia�
i:
a) f la�
phe�
p d�
�
i h�
nh
b) g kho�
ng pha�
i la�
phe�
p d�
�
i h�
nh ( v�
y1 �y2 th�M ��
N �MN )
6 Trong mpOxy cho phe�
p bie�
n h�
nh f : M(x;y) I��
� M�
=f(M) =(2x;y 1) . T�
m a�
nh cu�
a�
�
�
�
ng
tha�
ng () : x 3y 2 =0 qua phe�
p bie�
n h�
nh f .
Gia�
i:
Ca�
ch 1: Du�
ng bie�
u th�
�
c toa�
�
o�
-1-
� x�
�
�
x�
= 2x
x
Ta co�
f : M(x;y) I��
� M�
=f(M) = �
��
y�
y1 � 2
�
y y�
1
�
x�
V�M(x;y) �() � (
) 3(y�
1) 2 0 � x�
6y�
2 0 � M ���
(x ;y ) �(�
) : x 6y 2 0
2
Ca�
ch 2: La�
y 2�
ie�
m ba�
t k�M,N �() : M �N .
+M �() : M(2;0) I��
� M�
f(M) (4;1)
+N �( ) : N( 1; 1) I��
� N�
f(N) (2;0)
�
Qua M �
(4;1)
x+4 y 1
uuuuur
(�
) �(M ��
N ): �
� PTCta�
c (�
):
� PTTQ (�
): x 6y 2 0
��
6
1
VTCP
:
M
N
(6;
1
)
�
7 Trong mpOxy cho phe�
p bie�
n h�
nh f : M(x;y) I��
� M�
=f(M) =(x 3;y 1) .
a) CMR f la�
phe�
p d�
�
i h�
nh .
b) T�
m a�
nh cu�
a�
�
�
�
ng tro�
n (C) : (x +1)2 +(y 2)2 =4 .
I��
� (C�
) : (x 2)2 +(y 3)2 =4
8 Trong mpOxy cho phe�
p bie�
n h�
nh f : M(x;y) I��
� M�
=f(M) =(x 3;y 1) .
a) CMR f la�
phe�
p d�
�
i h�
nh .
b) T�
m a�
nh cu�
a�
�
�
�
ng tha�
ng ( ) : x +2y 5 =0 .
c) T�
m a�
nh cu�
a�
�
�
�
ng tro�
n (C) : (x +1)2 +(y 2)2 =2 .
x2
y2
+
=1 .
3
2
Gia�
i : a) La�
y hai �
ie�
m ba�
t k�M(x1;y1),N(x2;y2)
d ) T�
m a�
nh cu�
a elip (E) :
Khi �
o�
f : M(x1;y1) I��
� M�
=f(M) =(x1 3; y1 1) .
f : N(x2;y2) I��
� N�
=f(N) =(x2 3; y2 1)
Ta co�
: M ��
N = (x2 x1)2 (y2 y1)2 = MN
Va�
y : f la�
phe�
p d��
i h�
nh .
b) Ca�
ch 1: Du�
ng bie�
u th�
�
c toa�
�
o�
�
x�
=x 3 �
x x�
3
Ta co�
f : M(x;y) I��
� M�
=f(M) = �
��
�
�
y y1 �
y y 1
�
�
�
�
V�M(x;y) �() � (x 3) 2(y 1) 5 0 � x 2y�
4 0 � M ���
(x ;y ) �(�
) : x 2y 4 0
Ca�
ch 2: La�
y 2�
ie�
m ba�
t k�
M,N �() : M �N .
+M �() : M(5 ;0) I��
� M�
f(M) (2;1)
+N �() : N(3 ; 1) I��
� N�
f(N) (0;2)
-2-
�Qua M �
(2;1)
x 2 y1
uuuuur
(�
) �(M ��
N ): �
� PTCta�
c (�
):
� PTTQ(�
): x 2y 4 0
2
1
N (2;1)
�VTCP : M ��
Ca�
ch 3: V�f la�
phe�
p d�
�
i h�
nh ne�
n f bie�
n ���
�
ng tha�
ng () tha�
nh ��
��
ng tha�
ng (�
) // ( ) .
+La�
y M �() : M(5 ;0) I��
� M�
f(M) (2;1)
+V�(�
) // () � (�
): x +2y m =0 (m �5) . Do : (�
) M�
(2;1) � m = 4 � (�
): x 2y 4 0
c) Ca�
ch 1: Du�
ng bie�
u th��
c toa�
�o�
�x�
=x 3 �
x x�
3
Ta co�
f : M(x;y) I��
� M�
=f(M) = �
��
y1 �
y y�
1
�y�
V�M(x;y) �(C) : (x +1)2 +(y 2)2 =2 � (x�
4)2 (y�
3)2 2 �
� M ���
(x ;y ) �(C�
) : (x 4)2 (y 3)2 2
�
�
+Ta�
m I( 1;2) f
+Ta�
mI�
=f [I( 1;2)] (4;3)
Ca�
ch 2: (C) �
��
� (C�
)�
=R = 2
� BK : R = 2
� BK : R�
� (C�
) : (x 4)2 (y 3)2 2
d) Du�
ng bie�
u th�
�
c toa�
�
o�
�
x�
=x 3 �
x x�
3
Ta co�
f : M(x;y) I��
� M�
=f(M) = �
��
�
�
y y1 �
y y 1
�
V�
M(x;y) �(E) :
x2
y2
(x�
+3)2
(y�
1)2
(x +3)2 (y 1)2
+
=1 �
+
=1� M ���
(x ;y ) �(E�
):
+
=1
3
2
3
2
3
2
9 Trong mpOxy cho phe�
p bie�
n h�
nh f : M(x;y) I��
� M�
=f(M) =(x 1;y 2) .
a) CMR f la�
phe�
p d�
�
i h�
nh .
b) T�
m a�
nh cu�
a�
�
�
�
ng tha�
ng ( ) : x 2y 3 =0.
c) T�
m a�
nh cu�
a�
�
�
�
ng tro�
n (C) : (x +3)2 +(y 1)2 =2 .
d) T�
m a�
nh cu�
a parabol (P) : y2 =4x .
�S : b) x 2y 2 =0
c) (x +2)2 +(y 1)2 =2
d) (y +2)2 =4(x 1)
10 Trong mpOxy cho phe�
p bie�
n h�
nh f : M(x;y) I��
� M�
=f(M) =(x;y) . Kha�
ng �
�
nh na�
o sau �
a�
y
sai ?
A. f la�
1 phe�
p d�
�
i h�
nh
B. Ne�
u A(0 ; a) th�f(A) =A
C. M va�
f(M) �
o�
i x�
�
ng nhau qua tru�
c hoa�
nh
D. f [M(2;3)]��
�
�
�
ng tha�
ng 2x +y +1 =0
�S : Cho�
n C . V�
M va�
f(M) �
o�
i x�
�
ng nhau qua tru�
c tung � C sai .
12 Trong mpOxy cho 2 phe�
p bie�
n h�
nh :
f1 : M(x;y) I��
� M�
=f1(M) =(x +2 ; y 4) ; f2 : M(x;y) I��
� M�
=f2(M) =( x ; y) .
T�
m toa�
�
o�
a�
nh cu�
a A(4; 1) qua f1 ro�
i f2 , ngh�
a la�
t�
m f2[f1(A)] .
f
1� A �
2� A �
�
�S : A(4; 1) I��
(6; 5) I��
( 6; 5) .
x
11 Trong mpOxy cho phe�
p bie�
n h�
nh f : M(x;y) I��
� M�
=f(M) =( ; 3y) . Kha�
ng ��
nh na�
o sau �a�
y sai ?
2
A. f (O) =O (O la�
�ie�
m ba�
t bie�
n)
B. A�
nh cu�
a A �Ox th�a�
nh A �
=f(A) �Ox .
C. A�
nh cu�
a B �Oy th�a�
nh B�
=f(B) �Oy .
D. M �
=f [M(2 ; 3)] =(1; 9)
�S : Cho�
n D . V�
M�
=f [M(2 ; 3)] =(1; 9)
-3-
Vấn đề 2 : PHÉP TỊNH TIẾN
A. KIẾN THỨC CƠ BẢN
uuuuur r
r
1/ ĐN: Phép tịnh tiến theo véctơ ulà một phép dời hình biến điểm M thành điểm M �
sao cho MM �
u.
uuuuu
r r
r .Khi �
r (M) M �
K�hie�
u : T hay Tu
o�
: Tu
� MM �
u
gPhe�
p t�
nh tie�
n hoa�
n toa�
n�
�
�
�
c xa�
c�
�
nh khi bie�
t vect�t�
nh tie�
n cu�
a no�
.
r (M) M ,M th�Tr la�
gNe�
u To
p�
o�
ng nha�
t.
o phe�
r
r.
2/ Biểu thức tọa độ: Cho u =(a;b) và phép tịnh tiến Tu
�x�
=x +a
r (M) (x��
M(x;y) I��
� M�
=Tu
;y ) th��
�
�y =y +b
3/ Tính chất:
g�L : Phe�
p t�
nh tie�
n ba�
o toa�
n khoa�
ng ca�
ch gi��
a hai �
ie�
m ba�
t k�.
gHQ :
1. Ba�
o toa�
n t�
nh tha�
ng ha�
ng va�
th�
�
t��
cu�
a ca�
c�
ie�
m t�
�ng �
�
ng .
2. Bie�
n mo�
t tia tha�
nh tia .
3. Ba�
o toa�
n t�
nh tha�
ng ha�
ng va�
th��
t��
cu�
a ca�
c�
ie�
m t��
ng ��
ng .
5. Bie�
n mo�
t�
oa�
n tha�
ng tha�
nh �
oa�
n tha�
ng ba�
ng no�
.
6. Bie�
n mo�
t�
��
�
ng tha�
ng tha�
nh mo�
t�
��
�
ng tha�
ng song song hoa�
c tru�
ng v��
i�
�
��
ng tha�
ng �a�
cho .
7. Bie�
n tam gia�
c tha�
nh tam gia�
c ba�
ng no�
. (Tr�
�
c ta�
m I��
� tr�
�
c ta�
m , tro�
ng ta�
m I��
� tro�
ng ta�
m)
8. ��
�
�
ng tro�
n tha�
nh �
�
�
�
ng tro�
n ba�
ng no�
.
(Ta�
m bie�
n tha�
nh ta�
m : I I��
� I�
, R�
=R )
PHƯƠNG PHÁP TÌM ẢNH CỦA MỘT ĐIỂM
�x�
=x +a
r (M) (x��
M(x;y) I��
� M�
=Tu
;y ) th��
=y +b
�y�
PHƯƠNG PHÁP TÌM ẢNH CỦA MỘT HÌNH (H) .
Cách 1: Dùng tính chất (cùng phương của đường thẳng, bán kính đường tròn: không đổi)
1/ Lấy M ξ���
(H) I
2/
M � (H�
)
g(H) ��
�
�
�
ng tha�
ng ��
� (H�
) ��
�
�
�
ng tha�
ng cu�
ng ph�
�
ng
�
�
Ta�
mI
Ta�
m I�
g(H) �(C) �
I��
� (H�
) �(C�
)�
(ca�
n t�
mI�
).
+bk : R
+bk : R�
=R
�
�
Ca�
ch 2 : Du�
ng bie�
u th�
�
c to�
a�
o�
.
T�
m x theo x�
, t�
m y theo y�
ro�
i thay va�
o bie�
u th�
�
c to�
a�
o�
.
Ca�
ch 3 : La�
y hai �ie�
m pha�
n bie�
t : M, N ��
(H) I
M�
, N� (H�
)
B. BÀI TẬP
r
1 Trong mpOxy . T�
m a�
nh cu�
a M�
cu�
a�
ie�
m M(3; 2) qua phe�
p t�
nh tie�
n theo vect�u =(2;1) .
Gia�
i
uuuuu
r r
�
x�
3 2 �
x�
5
r (M) � MM �
Theo �
�
nh ngh�
a ta co�
: M�
=Tu
u � (x�
3;y�
2) (2;1) � �
��
y�
2 1 �
y�
1
�
� M�
(5; 1
-4-
r
2 T�
m a�
nh ca�
c�
ie�
m ch�
ra qua phe�
p t�
nh tie�
n theo vect�u :
r
a) A( 1;1) , u =(3;1)
r
b) B(2;1) , u =( 3;2)
r
c) C(3; 2) , u =( 1;3)
� A�
(2;3)
� B�
( 1;3)
� C�
(2;1)
r
3 Trong mpOxy . T�
m a�
nh A ��
,B la�
n l���
t cu�
a �ie�
m A(2;3), B(1;1) qua phe�
p t�
nh tie�
n theo vect�u =(3;1) .
uuur uuuur
T�
nh �o�
da�
i AB , A ��
B .
Gia�
i
uuur
uuuur
r (A) (5;4) , B�
r (B) (4;2) , AB =|AB | 5 , A ��
Ta co�
: A�
=Tu
=Tu
B =|A ��
B | 5 .
r r
r
r (M),M Tr (M ). T�
r (M) .
4 Cho 2 vect�u1;u2 . G�
a s��
M1 Tu
m v �e�
M 2 Tv
2
u
1
1
2
Gia�
i
uuuuur r
uuuuuuur r
r (M) � MM u
r (M ) � M M u
Theo �e�
: M1 Tu
,
M
T
1
1
2 u
1
1 2
2.
1 uuuuuu
r r r uuuuuur 2
uuuuur uuuuuuur r r
r r r
r (M) � MM v � v MM MM M M u +u .Va�
Ne�
u : M 2 Tv
y : v u1+u2
2
2
1
1 2 1 2
5 ��
�
�
ng tha�
ng ca�
t Ox ta�
i A( 1;0) , ca�
t Oy ta�
i B(0;2) . Ha�
y vie�
t ph�
�
ng tr�
nh �
�
�
�
ng tha�
ng �
la�
a�
nh
r
cu�
a qua phe�
p t�
nh tie�
n theo vect�u =(2; 1) .
r (A) (1; 1) , B�
r (B) (2;1) .
Gia�
i V�: A �
Tu
Tu
�
gqua A �
(1;uuuu
1)
r () � �
ur
Ma�
t kha�
c : �
Tu
�
i qua A ��
,B . Do �
o�
: �
�
��
g
VTCP
:
A
B
=(1;2)
�
�
x 1 t
� ptts �
:�
y 1 2t
�
6 ��
�
�
ng tha�
ng ca�
t Ox ta�
i A(1;0) , ca�
t Oy ta�
i B(0;3) . Ha�
y vie�
t ph�
�
ng tr�
nh �
�
�
�
ng tha�
ng �
la�
a�
nh
r
cu�
a qua phe�
p t�
nh tie�
n theo vect�u =( 1; 2) .
Gia�
i
r (A) (0; 2) , B�
r (B) (1;1) .
V�: A �
Tu
Tu
�
gqua A �
(0;uuuu
2)
�
x t
r () � �
ur
Ma�
t kha�
c : �
Tu
�
i qua A ��
,B . Do �
o�
: �
� ptts �
:�
�
y 2 3t
gVTCP : A �
B�
=( 1;3)
�
�
r
7 T�
�
ng t�
�
: a) : x 2y 4 =0 , u =(0 ; 3)
� �
: x 2y 2 0
r
b) : 3x y 3 =0 , u =( 1 ; 2)
� �
: 3x y 2 0
r
8 T�
m a�
nh cu�
a�
�
�
�
ng tro�
n (C) : (x +1)2 (y 2)2 4 qua phe�
p t�
nh tie�
n theo vect�u =(1; 3) .
Gia�
i
�
x�
=x +1 �
x =x�
1
r la�
Bie�
u th�
�
c toa�
�
o�
cu�
a phe�
p t�
nh tie�
n Tu
:�
��
�
�
y =y 3
y =y +3
�
�
2 (y�
V �: M(x;y) �(C) : (x +1)2 (y 2)2 4 � x�
1)2 4 � M ���
(x ;y )�(C�
) : x2 (y 1)2 4
Va�
y : A�
nh cu�
a (C) la�
(C�
) : x2 (y 1)2 4
9 Trong mpOxy cho phe�
p bie�
n h�
nh f : M(x;y) I��
� M�
=f(M) =(x 1;y 2) .
a) CMR f la�
phe�
p d�
�
i h�
nh .
b) T�
m a�
nh cu�
a�
�
�
�
ng tha�
ng ( ) : x 2y 3 =0.
c) T�
m a�
nh cu�
a�
�
�
�
ng tro�
n (C) : (x +3)2 +(y 1)2 =2 .
d) T�
m a�
nh cu�
a parabol (P) : y2 =4x .
�S : b) x 2y 2 =0
c) (x +2)2 +(y 1)2 =2
-5-
d) (y +2)2 =4(x 1)
10 Trong mpOxy cho phe�
p bie�
n h�
nh f : M(x;y) I��
� M�
=f(M) =(x;y) . Kha�
ng �
�
nh na�
o sau �
a�
y
sai ?
A. f la�
1 phe�
p d�
�
i h�
nh
B. Ne�
u A(0 ; a) th�f(A) =A
C. M va�
f(M) �
o�
i x�
�
ng nhau qua tru�
c hoa�
nh
D. f [M(2;3)]��
�
�
�
ng tha�
ng 2x +y +1 =0
�S : Cho�
n C . V�M va�
f(M) �
o�
i x�
�
ng nhau qua tru�
c tung � C sai .
r
9 T�
m a�
nh cu�
a�
�
�
�
ng tro�
n (C) : (x 3)2 (y 2)2 1 qua phe�
p t�
nh tie�
n theo vect�u =( 2;4) .
�
x�
=x 2 �x =x�
+2
r la�
Gia�
i : Bie�
u th�
�
c toa�
�
o�
cu�
a phe�
p t�
nh tie�
n Tu
:�
��
y�
=y 4 �y =y�
4
�
V�: M(x;y) �(C) : (x 3)2 (y 2)2 1� (x�
1)2 (y�
2)2 1� M ���
(x ;y ) �(C�
) : (x�
1)2 (y�
2)2 1
Va�
y : A�
nh cu�
a (C) la�
(C�
) : (x 1)2 (y 2)2 1
r
BT T�
�
ng t�
�
: a) (C) : (x 2)2 (y 3)2 1, u =(3;1)
r
b) (C) : x2 y2 2x 4y 4 0, u =( 2;3)
� (C�
) : (x 1)2 (y 2)2 1
(C�
) : x2 y2 2x 2y 7 0
10 Trong he�
tru�
c toa�
�
o�
Oxy , xa�
c�
�
nh toa�
�
o�
ca�
c�
�
nh C va�
D cu�
a h�
nh b�
nh ha�
nh ABCD bie�
t�
�
nh
A( 2;0), �
�
nh B( 1;0) va�
giao �
ie�
m ca�
c�
�
�
�
ng che�
o la�
I(1;2) .
Gia�
i
uur
uur
uur
gGo�
i C(x;y) .Ta co�
: IC (x 1;y 2),AI (3;2),BI (2; 1)
gV�
I la�
trung �
ie�
m cu�
a AC ne�
n:
uur uur
�
�
x 1 3
x 4
C =Tuur (I) � IC AI � �
��
� C(4;4)
AI
y 2 2 �
y 4
�
gV�
I la�
trung �
ie�
m cu�
a AC ne�
n:
uur uur
�
�
x 1 2
x 3
D =Tuur (I) � ID BI � � D
� �D
� D(3;4)
BI
yD 2 2 �
yD 4
�
Ba�
i ta�
p t�
�
ng t�
�
: A( 1;0),B(0;4),I(1;1)
� C(3;2),D(2; 2) .
. Hãy chỉ ra một phép tịnh tiến biến d thành d �
. Hỏi có bao
11 Cho 2 đường thẳng song song nhau d và d �
nhiêu phép tịnh tiến như thế?
Gia�
i : Cho�
n2�
ie�
m co�
��
nh A �d , A �
�d�
uuuuu
r uuur
La�
y �ie�
m tuy�
y�
M �d . G�
a s�
�
: M�
=Tuuur (M) � MM �
AB
AB
uuuu
r uuuur
� MA M �
B � M�
B / /MA � M �
�d�
� d�
=Tuuur (d)
AB
Nha�
n xe�
t : Co�
vo�
so�
phe�
p t�
nh tie�
n bie�
n d tha�
nh d�
.
12 Cho 2 �
�
�
�
ng tro�
n (I,R) va�
(I �
,R�
) .Ha�
y ch�ra mo�
t phe�
p t�
nh tie�
n bie�
n (I,R) tha�
nh (I �
,R�
).
uuuuu
r uu
r
u
u
r
Gia�
i : La�
y �ie�
m M tuy�
y�
tre�
n (I,R) . G�
a s�
�
: M�
=T (M) � M M �
II �
II �
uuu
r uuuu
r
r [(I,R)]
� IM I ��
M � I ��
M IM R � M �
�(I �
,R�
) � (I �
,R�
) =Tuu
II �
13 Cho h�
nh b�
nh ha�
nh ABCD , hai �
�
nh A,B co�
�
�
nh , ta�
m I thay �
o�
i di �
o�
ng
tre�
n�
�
�
�
ng tro�
n (C) .T�
m quy�
t�
ch trung �
ie�
m M cu�
a ca�
nh BC.
Gia�
i
uuu
r uur
Go�
i J la�
trung �
ie�
m ca�
nh AB . Khi �
o�
de�
tha�
y J co�
�
�
nh va�
IM JB .
Va�
y M la�
a�
nh cu�
a I qua phe�
p t�
nh tie�
n Tuur . Suy ra : Quy�
t�
ch cu�
a M la�
JB
uur
a�
nh cu�
a�
�
�
�
ng tro�
n (C) trong phe�
p t�
nh tie�
n theo vect�JB
-6-
r
14 Trong he�
tru�
c toa�
�
o�
Oxy , cho parabol (P) : y =ax2 . Go�
i T la�
phe�
p t�
nh tie�
n theo vect�u =(m,n)
va�
(P�
) la�
a�
nh cu�
a (P) qua phe�
p t�
nh tie�
n�
o�
. Ha�
y vie�
t ph�
�
ng tr�
nh cu�
a (P�
).
Gia�
i:
uuuuu
r r
uuuuu
r
r
Tu
gM(x;y) I���
� M ���
(x ;y ) , ta co�
: MM �
=u , v��
i MM �
=(x�
x ; y�
y)
uuuuu
r r
�
x�
x =m �
x =x�
m
V�MM �
=u � �
��
y�
y =n
y =y�
n
�
�
Ma�
: M(x;y) �(P): y ax2 � y�
n =a(x�
m)2 � y�
=a(x�
m)2 n � M ���
(x ;y ) �(P�
) : y =a(x m)2 n
r la�
Va�
y : A�
nh cu�
a (P) qua phe�
p t�
nh tie�
n Tu
(P�
) : y =a(x m)2 n � y =ax2 2amx am2 n .
r r
r ( ) .
15 Cho �
t : 6x +2y 1=0 . T�
m vect�u �0 �
e�
=Tu
r
r
r
r
r ( ) � u
Gia�
i : VTCP cu�
a la�
a =(2; 6) . �e�
: =Tu
cu�
ng ph�
�
ng a . Khi �
o�
: a =(2; 6) 2(1; 3)
r
� cho�
n u =(1; 3) .
r
r
r (A) , C =Tr (B) . T�
16 Trong he�
tru�
c toa�
�
o�
Oxy , cho 2 �ie�
m A( 5;2) , C( 1;0) . Bie�
t : B =Tu
m u va�
v
v
�
e�
co�
the�
th�
�
c hie�
n phe�
p bie�
n�
o�
i A tha�
nh C ?
Gia�
i
uuur r uuur r uuur uuur uuur r r
r
Tu
Tvr
A( 5;2) I���
� B I��
�
� C(1;0) . Ta coù: AB u,BC v � AC AB BC u v (4; 2)
r r
Tu+v
r
r
17 Trong he�
tru�
c toa�
�
o�
Oxy , cho 3 �
ie�
m K(1;2) , M(3; 1),N(2; 3) va�
2 vect�u =(2;3) ,v =( 1;2) .
r ro�
r.
T�
m a�
nh cu�
a K,M,N qua phe�
p t�
nh tie�
n Tu
i Tv
uuur r uuur r uuur uuur uuur r r
r
r
Tu
Tv
HD : G�
a s�
�
: A(x;y) I���
� B I���
� C(x��
;y ) . Ta co�
: AB u,BC v � AC AB BC u v (1;5)
uuuu
r
�
�
x�
1 1
x�
2
r r (K) � KK �
Do �
o�
: K�
=Tu
(1;5) � �
��
� K�
(2;7) .
v
y�
2 5 �
y�
7
�
T�
�
ng t�
�
: M�
(4;4) , N�
(3;2) .
18 Trong he�
tru�
c toa�
�
o�
Oxy , cho ABC : A(3;0) , B( 2;4) , C( 4;5) . G la�
tro�
ng ta�
m ABC va�
phe�
p
r r
r (G) .
t�
nh tie�
n theo vect�u �0 bie�
n A tha�
nh G . T�
m G�
=Tu
Gia�
i
r
r
Tu
Tu
A(3;0) I���
� G(1;3) I���
� G���
(x ;y )
uuur
u
u
u
u
r
r
r
�
x�
1 4 �
x�
5
V�AG (4;3) u . Theo �
e�
: GG�
u� �
��
� G�
(5;6).
y�
3 3
y�
6
�
�
19 Trong ma�
t pha�
ng Oxy , cho 2 �
��
�
ng tro�
n (C) : (x 1)2 (y 3)2 2,(C�
): x2 y2 10x 4y 25 0.
r
Co�
hay kho�
ng phe�
p t�
nh tie�
n vect�u bie�
n (C) tha�
nh (C�
).
HD : (C) co�
ta�
m I(1; 3), ba�
n k�
nh R =2 ; (C�
) co�
ta�
m I�
(5; 2), ba�
n k�
nh R�
=2 .
r
Ta tha�
y : R =R�
=2 ne�
n co�
phe�
p t�
nh tie�
n theo vect�u =(4;1) bie�
n (C) tha�
nh (C�
).
20 Trong he�
tru�
c toa�
�
o�
Oxy , cho h�
nh b�
nh ha�
nh OABC v��
i A( 2;1) va�
B � :2x y 5 =0 . T�
m ta�
p
h�
�
p�
�
nh C ?
Gia�
i
uuur uuur
r
r (B) v��
gV�OABC la�
h�
nh b�
nh ha�
nh ne�
n : BC AO (2; 1) � C Tu
i u =(2; 1)
uuur r
r
Tu
�
�
x�
x 2
x x�
2
gB(x;y) I���
� C(x��
;y ) . Do : BC u � �
��
�
�
y y 1 �
y y 1
�
gB(x;y) � � 2x y 5 =0 � 2x�
y�
10 =0 � C(x��
; y ) ��
:2x y 10 =0
-7-
21 Cho ABC . Go�
i A1,B1,C1 la�
n l�
�
�
t la�
trung �
ie�
m ca�
c ca�
nh BC,CA,AB. Go�
i O1,O2,O3 va�
I1,I 2,I 3
t�
�
ng �
�
ng la�
ca�
c ta�
m�
�
�
�
ng tro�
n ngoa�
i tie�
p va�
ca�
c ta�
m�
�
�
�
ng tro�
n no�
i tie�
p cu�
a ba tam gia�
c AB1C1,
BC1A1, va�
CA1B1 . Ch�
�
ng minh ra�
ng : O1O2O3 I1I 2I 3 .
HD :
wXe�
t phe�
p t�
nh tie�
n : T1uuur bie�
n A I��
� C,C1 I��
� B,B1 I��
� A1 .
AB
2
T1uuur
T1uuur
T1uuur
AB
AB
AB
2 � C BA ;O I����
2 � O ;I I����
2 �I .
� AB1C1 I����
1 1 1
2 1
2
uuuuuur uuuur
� O1O2 I1I 2 � O1O2 I1I 2.
wLy�
lua�
n t�
�
ng t�
�
: Xe�
t ca�
c phe�
p t�
nh tie�
n T1uuur ,T1uuur suy ra :
BC
CA
2
2
uuuuuur uuuur
uuuuuu
r uuuu
r
O2O3 I 2I 3 va�
O3O1 I 3I1 � O2O3 I 2I 3,O3O1 I 3I1 � O1O2O3 I1I 2I 3 (c.c.c).
� 60o,B
� 150ova�
� 90o.
22 Trong t�
�
gia�
c ABCD co�
AB =6 3cm ,CD 12cm , A
D
T�
nh �
o�
da�
i ca�
c ca�
nh BC va�
DA .
HD :
uuuu
r uuur
Tuuur
�
� 150o)
BC � M � AM BC.Ta co�
wXe�
t : A I���
: ABCM la�
h�
nh b�
nh ha�
nh va�
BCM
30o(v�B
� 360o (90o 60o 150o) 60o � MCD
� 30o.
La�
i co�
: BCD
��
nh ly�
ha�
m cos trong MCD :
3
MD2 MC2 DC2 2MC.DC.cos30o (6 3)2 (12)2 2.6 3.12.
36
2
� MD =6cm .
1
Ta co�
: MD = CD va�
MC =MD 3 � MDC la�
tam gia�
c�
e�
u
2
� 90o va�
�
� MCD la�
n�
�
a tam gia�
c�
e�
u � DMC
MDA
30o.
�
�
� 30o � AMD la�
Va�
y : MDA
MAD
MAB
tam gia�
c ca�
n ta�
iM.
6 3
Dựng MK AD � K làtrung điể
m củ
a AD � KD=MDcos30o
cm � AD 6 3cm
2
Tó
m lại : BC =AM =MD =6cm , AD =AB =6 3cm
-8-
Vn 3 : PHẫP I XNG TRC
A . KIN THC C BN
1/ N1:im M
gi l i xng vi im M qua ng thng a nu a l ng trung trc ca on MM
Phe
p
o
i x
ng qua
ng tha
ng co
n go
i la
phe
p
o
i x
ng tru
c .
ng tha
ng a go
i la
tru
c
o
i x
ng.
N2 : Phe
p
o
i x
ng qua
ng tha
ng a la
phe
p bie
n h
nh bie
n mo
i
ie
m M tha
nh
ie
mM
o
i x
ng
v
i M qua
ng tha
uuuunug
ura . uuuuuu
r
Khie
u : a(M) M
M oM
M oM , v
i M o la
h
nh chie
u cu
a M tre
n
ng tha
ng a .
Khi ú :
gNe
u M a tha(M) M : xem M la
o
i x
ng v
i ch
nh no
qua a . ( M co
n go
i la
ie
m ba
t
o
ng )
gM a tha(M) M
a la
ng trung tr
c cu
a MM
gẹa(M) M
thỡ ẹa(M
) M
gẹa(H) H
thỡ ẹa(H
) H , H
laứaỷ
nh cuỷ
a hỡnh H .
gN : d la
tru
c
o
i x
ng cu
a h
nh H d(H) H .
gPhe
p
o
i x
ng tru
c hoa
n toa
n xa
c
nh khi bie
t tru
c
o
i x
ng cu
a no
.
Chu
y
: Mo
t h
nh co
the
kho
ng co
tru
c
o
i x
ng ,co
the
co
mo
t hay nhie
u tru
c
o
i x
ng .
2/ Biu thc ta : M(x;y) I
M
d(M) (x
;y )
x
x
=x
= x
d Ox :
d Oy :
y
=
y
y
=y
3/ L: Phộp i xng trc l mt phộp di hỡnh.
gHQ :
1.Phe
p
o
i x
ng tru
c bie
n ba
ie
m tha
ng ha
ng tha
nh ba
ie
m tha
ng ha
ng va
ba
o toa
n th
t
cu
a ca
c
ie
m t
ng
ng .
2.
ng tha
ng tha
nh
ng tha
ng .
3. Tia tha
nh tia .
4. oa
n tha
ng tha
nh
oa
n tha
ng ba
ng no
.
5. Tam gia
c tha
nh tam gia
c ba
ng no
. (Tr
c ta
mI
tr
c ta
m , tro
ng ta
mI
tro
ng ta
m)
6.
ng tro
n tha
nh
ng tro
n ba
ng no
. (Ta
m bie
n tha
nh ta
m : I I
I
, R
=R )
7. Go
c tha
nh go
c ba
ng no
.
PP : Tỡm aỷ
nh M
=ẹa(M)
1. (d) M , d a
2. H =d a
3. H laứtrung ủieồ
m cuỷ
a MM
M
?
PP : T
m a
nh cu
a
ng tha
ng :
=a()
wTH1:()// (a)
1. La
y A,B () : A B
2. T
m a
nh A
=a(A)
3.
A
, // (a)
w TH2 : // a
1. T
m K = a
2. La
y P : P K .T
m Q =a(P)
3.
(KQ)
m M () : (MA +MB)min.
PP : T
-9-
T�
m M �() : (MA+MB)min
wLoa�
i 1 : A, B na�
m cu�
ng ph�
a�
o�
i v�
�
i () :
�
1) go�
i A la�
�o�
i x�
�
ng cu�
a A qua ()
2) M �(), th�MA +MB MA �
+MB �A �
B
�
�
Do �
o�
: (MA+MB)min=A B � M =(A B) �()
wLoa�
i 2 : A, B na�
m kha�
c ph�
a�
o�
i v��
i () :
M �( ), th�MA +MB �AB
Ta co�
: (MA+MB)min =AB � M =(AB) �()
B . BÀI TẬP
1 Trong mpOxy . T�
m a�
nh cu�
a M(2;1) �
o�
i x�
�
ng qua Ox , ro�
i�
o�
i x�
�
ng qua Oy .
�
�
Oy
Ox � M �
�
HD : M(2;1) I���
(2; 1) I���
� M�
(2; 1)
2 Trong mpOxy . T�
m a�
nh cu�
a M(a;b) �
o�
i x�
�
ng qua Oy , ro�
i�
o�
i x�
�
ng qua Ox .
�
�
Oy
Ox � M �
�
HD : M(a;b) I���
� M�
( a;b) I���
(a; b)
�
�
a�
�
3 Cho 2 �
�
�
�
ng tha�
ng (a) : x 2 =0 , (b) : y +1 =0 va�
�
ie�
m M( 1;2) . T�
m : M I��
� M�
I��b�
� M�
.
�
�
a�
�
HD : M( 1;2) I��
� M�
(5;2) I��b�
� M�
(5; 4) [ ve�
h�
nh ] .
4 Cho 2 �
�
�
�
ng tha�
ng (a) : x m =0 (m >0) , (b) : y +n =0 (n >0).
�
�
a�
b� M �
�
��
T�
m M�
: M(x;y) ��
� M ���
(x ;y ) ���
(x ��
;y �
).
�
�a
�b
�
�
x�
2m x
x�
2m x
�
HD : M(x;y) I����
� M�
I������
� M�
�
�
t�
(m;y)
t�(2mx; n)
�
y�
y
y�
2n y
�
�
5 Cho �
ie�
m M( 1;2) va�
�
�
�
�
ng tha�
ng (a) : x +2y +2 =0 .
HD : (d) : 2x Ǯ
y+
�4
=0 , H =d a H( 2;0) , H la�
trung �
ie�
m cu�
a MM � M �
( 3; 2)
6 Cho �
ie�
m M( 4;1) va�
�
�
�
�
ng tha�
ng (a) : x +y =0 .
� M�
=�a(M) (1;4)
7 Cho 2 �
�
�
�
ng tha�
ng ( ) : 4x y +9 =0 , (a) : x y +3 =0 . T�
m a�
nh �
=�a( ) .
HD :
4 1
gV� � ��
ca�
t aǮ K
a K( 2;1)
1 1
gM( 1;5) � � d M, a � d: x y 4 0 � H(1/ 2;7/ 2): t�
ie�
m cu�
a MM �
� M�
�a(M) (2;2)
g�
�KM �
: x 4y +6 =0
8 T�
m b =�a(Ox) v�
�
i�
�
�
�
ng tha�
ng (a) : x +3y +3 =0 .
HD : ga�Ox =K( 3;0) .
3 9
gM �O(0;0) �Ox : M �
=�a(M) =( ; ) .
5 5
gb �KM �
: 3x +4y 9 =0 .
9 T�
m b =�a(Ox) v�
�
i�
�
�
�
ng tha�
ng (a) : x +3y 3 =0 .
- 10 -
HD : ga�Ox =K(3;0) .
gP �O(0;0) �Ox .
�
+Qua O(0;0)
g �
� :3x y 0
+a
�
3 9
3 9
gE =a� � E( ; ) la�
trung �
ie�
m OQ � Q( ; ) .
10 10
5 5
gb �KQ : 3x +4y 9 =0 .
10 T�
m b =�Ox(a) v��
i�
��
�
ng tha�
ng (a) : x +3y 3 =0 .
Gia�
i:
Ca�
ch 1: Du�
ng bie�
u th�
�
c toa�
�
o�(ra�
t hay)
Ca�
ch 2: gK=aǮ Ox K(3;0)
gP(0;1) �a � Q =�Ox (P) =(0; 1)
gb �KQ : x 3y 3 =0 .
11 Cho 2 �
�
�
�
ng tha�
ng ( ) : x 2y +2 =0 , (a) : x 2y 3 =0 . T�
m a�
nh �
=�a( ) .
PP : / /a
Ca�
ch 1 : T�
m A,B � � A ��
,B ��
� �
�A �
B�
Ca�
ch 2 : T�
m A � � A �
��
� �
/ / , �
A�
Gia�
i : gA(0;1) � � A �
�a(A) (2; 3)
g�
A ��
, / / � �
: x 2y 8 0
12 Cho �
�
�
�
ng tro�
n (C) : (x+3)2 (y 2)2 1 , �
�
�
�
ng tha�
ng (a) : 3x y +1=0 . T�
m (C�
) =�a[(C)]
HD : (C�
) : (x 3)2 y2 1 .
13 Trong mpOxy cho ABC : A( 1;6),B(0;1) va�
C(1;6) . Kha�
ng �
�
nh na�
o sau �
a�
y sai ?
A. ABC ca�
n�
�
B
B. ABC co�
1 tru�
c�
o�
i x�
�
ng
C. ABC �Ox (ABC)
D. Tro�
ng ta�
m : G =�Oy (G)
HD : Cho�
nD
14 Trong mpOxy cho �ie�
m M( 3;2), ����
ng tha�
ng () : x +3y 8 =0, ����
ng tro�
n (C) : (x+3)2 (y 2)2 4.
T�
m a�
nh cu�
a M, () va�
(C) qua phe�
p �o�
i x��
ng tru�
c (a) : x 2y +2 =0 .
Gia�
i : Go�
i M�
, (�
) va�
(C�
) la�
a�
nh cu�
a M, () va�
(C) qua phe�
p �o�
i x��
ng tru�
c a.
�
g Qua M( 3;2)
a) T�
m a�
nh M �
: Go�
i ����
ng tha�
ng (d) :�
ga
�
+(d) (a) � (d) : 2x y +m =0 . V�(d) M( 3;2) � m = 4 � (d): 2x y 4 =0
�
1
xH (xM xM �
)
�
2
+H =(d) �(a) � H( 2;0) � H la�
trung �
ie�
m cu�
a M,M �
� H�
1
�
yH (yM yM �
)
�
2
�
1
2 (3 xM �
)
�
�
x
1
2
��
� �M�
� M�
(1; 2)
1
y
2
�
�
M
�
0 (2 yM �
)
� 2
b) T�
m a�
nh (�
):
1
3
gV� � � ( ) ca�
t (a) � K =( ) �(a)
1 2
�
x +3y 8 =0
� Toa�
�
o�
cu�
a K la�
nghie�
m cu�
a he�
: �
� K (2;2)
x 2y +2 =0
�
- 11 -
gLa�
y P �K � Q =�a[P( 1;3)] =(1; 1) . ( La�
m t�
�
ng t�
�
nh�ca�
u a) )
�
g Qua P( 1;3)
Go�
i �
�
�
�
ng tha�
ng (b) : �
g a
�
+(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 �
ie�
m cu�
a P,Q �
�
� 1
1
xE (xP xQ ) �
0 (1 xQ ) �
xQ 1
�
�
� 2
�
2
� E�
��
��
� Q(1; 1)
1
1
yQ 1
�
�
�
y (y yQ )
1 (3 yQ )
�E 2 P
� 2
�
gQua K(2;2)
x 2 y 2
uuur
+(�
) �(KQ) : �
� (�
):
� 3x y 4 0
1
3
gVTCP : KQ (1; 3) (1;3)
�
c) +T�
m a�
nh cu�
a ta�
m I( 3;2) nh�ca�
u a) .
�a
�a
m I I���
m I � .T�
+ V�phe�
p�
o�
i x�
�
ng tru�
c la�
phe�
p d�
�
i h�
nh ne�
n (C): gTa�
�(C�
): gTa�
m I I���
�I�
gR 2
gR�
R2
�
2 2
�a
�
m I( 3;2)I���
+Ta�
m I�
=�a [I( 3; 2)] ( ; )
Va�
y : (C) +Ta�
� (C�
)�
5 5
BK : R =2
�
�
BK
:
R
=
R
=
2
�
22
22
� (C�
) : (x ) (y ) 4
5
5
15 Trong mpOxy cho �ie�
m M(3; 5), ����
ng tha�
ng () : 3x +2y 6 =0, ����
ng tro�
n (C) : (x+1)2 (y 2)2 9.
T�
m a�
nh cu�
a M, () va�
(C) qua phe�
p �o�
i x��
ng tru�
c (a) : 2x y +1 =0 .
HD :
�a
33 1
9 13
a) M(3; 5) I���
� M�
( ; ),(d): x 2y 7 0,t�ie�
m H( ; )
5 5
5 5
4 15
b) +K= Ǯ (a) K( ; )
7 7
+P �() : P(2;0) �K , Q =�a[P(2;0)] =( 2;2)
� (�
) �(KQ) : x 18y 38 0
�a
9 8
9
8
c) +I(1; 2) I���
� I�
( ; ) , R�
=R =3
� (C�
) : (x + )2 (y )2 9
5 5
5
5
16 Cho �
ie�
m M(2; 3), �
�
�
�
ng tha�
ng () : 2x +y 4 =0, �
�
�
�
ng tro�
n (C) : x2 y2 2x 4y 2 0.
T�
m a�
nh cu�
a M, () va�
(C) qua phe�
p�
o�
i x�
�
ng qua Ox .
�Ox
�x x�
�x�
x
HD : Ta co�
: M(x;y) ���
� M�
(1) � �
(2)
�
y
�y y�
�y�
�
Ox � M �
gThay va�
o (2) : M(2; 3) ���
(2;3)
gM(x;y) �() � 2x�
y�
4 =0 � M ���
(x ;y ) �(�
) : 2x y 4 =0 .
2 y�
2 2x�
gM(x;y) �(C) : x2 y2 2x 4y 2 0 � x�
4y�
2 0
� (x�
1)2 (y�
2)2 3 � M ���
(x ;y ) �(C�
) : (x 1)2 (y 2)2 3
17 Trong mpOxy cho �
�
�
�
ng tha�
ng (a) : 2x y+3 =0 . T�
m a�
nh cu�
a a qua �Ox.
�Ox
�
�
x�
x
x x�
Gia�
i : Ta co�
: M(x;y) I���
� M�
��
��
y y �
y y�
�
�: 2x y +3 =0
V�
M(x;y) �(a) : 2x y+3 =0 � 2(x�
) (y�
)+3 =0 � 2x�
y�
+3 =0 � M �
(x��
;y ) �(a)
�
Oy
�: 2x y +3 =0
Va�
y : (a) I���� (a)
- 12 -
18 Trong mpOxy cho �
�
�
�
ng tro�
n (C) : x2 y2 4y 5 =0 . T�
m a�
nh cu�
a a qua �Oy.
�Oy
�
x�
x �
x x�
Gia�
i : Ta co�
: M(x;y) I���� M �
��
�
�
y
y
y y�
�
�
2 4(y�
2 y�
2 4y 5 =0
V�M(x;y) �(C) : x2 y2 4y 5 =0 � ( x�
)2 y�
) 5 =0 � x�
� M ���
(x ;y ) �(C�
) : x2 y2 4y 5 =0
�Oy
Va�
y : (C) I���� (C�
) : x2 y2 4y 5 =0
19 Trong mpOxy cho �
tha�
ng (a) : 2x y 3 =0 , () : x 3y 11 =0 , (C) : x2 y2 10x 4y 27 =0 .
a) Vie�
t bie�
u th�
�
c gia�
i t�
ch cu�
a phe�
p�
o�
i x�
�
ng tru�
c �a .
b) T�
m a�
nh cu�
a�
ie�
m M(4; 1) qua �a.
c) T�
m a�
nh : (�
) =�a(),(C�
) �a(C) .
Gia�
i
a) To�
ng qua�
t (a) : Ax +By +C=0 , A 2 B2 �0
uuuuu
r
uuuuu
r r
�a
r
Go�
i M(x;y) I���
� M ���
(x ;y ) , ta co�
: MM �
(x�
x;y�
y) cu�
ng ph�
�
ng VTPT n =(A;B) � MM �
tn
x x�y y�
�
x�
x At �
x�
x At
��
��
(t ��) . Go�
i I la�
trung �
ie�
m cu�
a MM �
ne�
n I(
;
) �(a)
y�
y Bt �
y�
y Bt
2
2
�
x x�
y y�
x x At
y y Bt
� A(
) B(
) C 0 � A(
) B(
) C 0
2
2
2
2
2(Ax +By +C)
� (A 2 B2)t 2(Ax +By +C) � t
A 2 B2
�
2A(Ax +By +C)
2B(Ax +By +C)
��
x�
x
;y�
y
�
A 2 B2
A 2 B2
�
�
4(2x y 3)
3
4 12
x�
x
x�
x y
�
�
�
�
5
5
5
5
A�
p du�
ng ke�
t qua�
tre�
n ta co�
:�
��
2(2x
y
3)
4
3
6
�
�
y�
y
y�
y y
5
5
5
�
� 5
�a
4 7
b) M(4; 1) I���
� M�
( ; )
5 5
�
a� �
c) I���
:3x y 17 0
�
a� (C�
d) (C) I���
):(x 1)2 (y 4)2 2
20 Trong mpOxy cho đườ
ng thẳ
ng ( ) : x 5y 7 =0 và(�
) : 5x y 13 =0 . Tìm phé
p đố
i xứ
ng qua
�
trục biế
n ( ) thà
nh ( ) .
Giả
i
1 5
Vì � � () và(�
) cắ
t nhau . Do đótrục đố
i xứ
ng (a) củ
a phé
p đố
i xứ
ng biế
n ( ) thà
nh (�
) chính
5 1
làđườ
ng phâ
n giá
c củ
a gó
c tạo bở
i () và(�
).
x y 5 0 (a1)
| x 5y 7| |5x y 13| �
��
x y 1 0 (a2)
1 25
25 +1
�
Vậ
y có2 phé
p đố
i xứ
ng qua cá
c trục (1) : x y 5 0 , ( 2): x y 1 0
Từđósuy ra (a) :
21 Qua phe�
p�
o�
i x�
�
ng tru�
c �a :
1. Nh�
�
ng tam gia�
c na�
o bie�
n tha�
nh ch�
nh no�
?
2. Nh�
�
ng �
�
�
�
ng tro�
n na�
o bie�
n tha�
nh ch�
nh no�
?
- 13 -
HD :
1. Tam gia�
c co�
1�
�
nh �tru�
c a , hai �
�
nh co�
n la�
i�
o�
i x�
�
ng qua tru�
ca.
2. ��
�
�
ng tro�
n co�
ta�
m �a .
22 T�
m a�
nh cu�
a�
�
�
�
ng tro�
n (C) : (x 1)2 (y 2)2 4 qua phe�
p�
o�
i x�
�
ng tru�
c Oy.
PP : Du�
ng bie�
u th�
�
c toa�
�
o�� �S : (C�
) : (x 1)2 (y 2)2 4
23 Hai ABC va�
A �
B��
C cu�
ng na�
m trong ma�
t pha�
ng toa�
�
o�
va�
�
o�
i x�
�
ng nhau qua tru�
c Oy .
Bie�
t A( 1;5),B(4;6),C�
(3;1) . Ha�
y t�
m toa�
�
o�
ca�
c�
�
nh A �
, B�
va�
C.
�S : A �
(1;5), B�
(4;6) va�
C( 3;1)
24 Xe�
t ca�
c h�
nh vuo�
ng , ngu�
gia�
c�
e�
u va�
lu�
c gia�
c�
e�
u . Cho bie�
t so�
tru�
c�
o�
i x�
�
ng t�
�
ng �
�
ng cu�
a mo�
i
loa�
i�
a gia�
c�
e�
u�
o�
va�
ch�ra ca�
ch ve�
ca�
c tru�
c�
o�
i x�
�
ng �
o�
.
�S :
gH�
nh vuo�
ng co�
4 tru�
c�
o�
i x�
�
ng , �
o�
la�
ca�
c�
�
�
�
ng tha�
ng �
i qua 2 �
�
nh �
o�
i die�
n va�
ca�
c�
�
�
�
ng tha�
ng
�
i qua trung �
ie�
m cu�
a ca�
c ca�
p ca�
nh �
o�
i die�
n.
gNgu�
gia�
c�
e�
u co�
5 tru�
c�
o�
i x�
�
ng ,�
o�
la�
ca�
c�
�
�
�
ng tha�
ng �
i qua �
�
nh �
o�
i die�
n va�
ta�
m cu�
a ngu�
gia�
c�
e�
u.
gLu�
c gia�
c�
e�
u co�
6 tru�
c�
o�
i x�
�
ng , �
o�
la�
ca�
c�
�
�
�
ng tha�
ng �
i qua 2 �
�
nh �
o�
i die�
n va�
ca�
c�
�
�
�
ng tha�
ng �
i
qua trung �
ie�
m cu�
a ca�
c ca�
p ca�
nh �
o�
i die�
n.
25 Go�
i d la�
pha�
n gia�
c trong ta�
i A cu�
a ABC , B�la�
a�
nh cu�
a B qua phe�
p�
o�
i x��
ng tru�
c �d . Kha�
ng �
�
nh
na�
o sau �a�
y sai ?
A. Ne�
u AB
�
tre�
n ca�
nh AC .
B. B�
la�
trung �
ie�
m ca�
nh AC .
C. Ne�
u AB =AC th�B�
�C .
D. Ne�
u B�la�
trung �ie�
m ca�
nh AC th�AC =2AB .
�S : Ne�
u B�
=�d(B) th�B�
�AC .
gA �
u�
ng . V�AB
AB�
=AB ne�
n AB�
�
�
tre�
n ca�
nh AC .
1
gB sai . V�gia�
thie�
t ba�
i toa�
n kho�
ng �u�
kha�
ng �
�
nh AB = AC.
2
�
�
�
gC �
u�
ng . V�AB =AB ma�
AB =AC ne�
n AB =AC
B C .
�
gD �
u�
ng . V�Ne�
u B la�
trung �
ie�
m ca�
nh AC th�AC=2AB�
ma�
AB�
=AB ne�
n AC=2AB .
26 Cho 2 �
�
��
ng tha�
ng a va�
b ca�
t nhau ta�
i O . Xe�
t 2 phe�
p�
o�
i x�
�
ng tru�
c �a va�
�b :
�
�
a� B I���
b� C . Kha�
A I���
ng ��
nh na�
o sau �a�
y kho�
ng sai ?
A. A,B,C ����
�
ng tro�
n (O, R =OC) .
B. T��
gia�
c OABC no�
i tie�
p.
C. ABC ca�
n�
�
B
D. ABC vuo�
ng ��
B
HD : gA. Kho�
ng sai . V�d1 la�
trung tr�
�
c cu�
a AB � OA =OB , d2 la�
trung tr�
�
c
cu�
a BC � OB =OC � OA =OB =OC � A,B,C ���
��
ng tro�
n (O, R =OC) .
gCa�
c ca�
u B,C,D co�
the�
sai .
27 Cho ABC co�
hai tru�
c�
o�
i x�
�
ng . Kha�
ng �
�
nh na�
o sau �
a�
y�
u�
ng ?
A. ABC la�
vuo�
ng
B. ABC la�
vuo�
ng ca�
n
C. ABC la�
�
e�
u
HD : G�
a s�
�
ABC co�
2tru�
c�
o�
i x�
�
ng la�
AC va�
BC
�
AB =AC
��
� AB AB BC � ABC �
e�
u.
BC =BA
�
- 14 -
D. ABC la�
ca�
n.
� 110o. T�
� va�
��
28 Cho ABC co�
A
nh B
C
e�
ABC
co�
tru�
c�
o�
i x�
�
ng .
� =50o va�
� 20o
� =45o va�
� 25o
A. B
C
B. B
C
� =40o va�
� 30o
C. B
C
HD : Chọn D . Vì : ABC cótrục đố
i xứ
ng khi ABC câ
n hoặ
c đề
u
� 110o 90o � ABC câ
Vì A
n tại A , khi đó:
� =C
� 35o
D. B
� 180o 110o
180o A
�
�
BC
35o
2
2
29 Trong ca�
c h�
nh sau , h�
nh na�
o co�
nhie�
u tru�
c�
o�
i x�
�
ng nha�
t?
A. H�
nh ch�
�
nha�
t
B. H�
nh vuo�
ng
C. H�
nh thoi
�S : Cho�
n B. V�: H�
nh vuo�
ng co�
4 tru�
c�
o�
i x�
�
ng .
D. H�
nh thang ca�
n.
30 Trong ca�
c h�
nh sau , h�
nh na�
o co�
�
t tru�
c�
o�
i x�
�
ng nha�
t?
A. H�
nh ch�
�
nha�
t
B. H�
nh vuo�
ng
C. H�
nh thoi
�S : Cho�
n D. V�: H�
nh thang ca�
n co�
1 tru�
c�
o�
i x�
�
ng .
31 Trong ca�
c h�
nh sau , h�
nh na�
o co�
3 tru�
c�
o�
i x�
�
ng ?
A. H�
nh thoi
B. H�
nh vuo�
ng
�S : Cho�
n C. V�
: �
e�
u co�
3 tru�
c �o�
i x�
�
ng .
C. �
e�
u
D. H�
nh thang ca�
n.
D. vuo�
ng ca�
n.
32 Trong ca�
c h�
nh sau , h�
nh na�
o co�
nhie�
u h�
n 4 tru�
c�
o�
i x�
�
ng ?
A. H�
nh vuo�
ng
B. H�
nh thoi
C. H�
nh tro�
n
�S : Cho�
n C. V�
: H�
nh tro�
n co�
vo�
so�
tru�
c�
o�
i x�
�
ng .
D. H�
nh thang ca�
n.
33 Trong ca�
c h�
nh sau , h�
nh na�
o kho�
ng co�
tru�
c�
o�
i x�
�
ng ?
A. H�
nh b�
nh ha�
nh
B. �
e�
u
C. ca�
n
D. H�
nh thoi .
�S : Cho�
n A. V�: H�
nh b�
nh ha�
nh kho�
ng co�
tru�
c�
o�
i x�
�
ng .
34 Cho hai h�
nh vuo�
ng ABCD va�
AB���
C D co�
ca�
nh �
e�
u ba�
ng a va�
co�
�
�
nh A chung .
Ch�
�
ng minh : Co�
the�
th�
�
c hie�
n mo�
t phe�
p�
o�
i x�
�
ng tru�
c bie�
n h�
nh vuo�
ng ABCD tha�
nh�
AB���
CD .
��
HD : G�
a s�
�
: BC �B C =E .
�B
��
Ta co�
: AB =AB�
,B
90o,AE chung .
�ABE
���
= AB
�
F
�
�
EB =EB�
B I AE
�
t AB =AB�
�bie�
B�
�
�
EC =EC�
Ma�
t kha�
c: �
���
C I AE C�
AC =AC�
=a 2
�
�
��
� 90o BAB�
Ngoa�
i ra : AD�
=AD va�
D
AE DAE
2
�A
�AE
����
D I ��
D� ABCD I
AB���
CD
35 Gọi H làtrực tâ
m ABC . CMR : Bố
n tam giá
c ABC , HBC , HAC , HAC có
đườ
ng trò
n ngoại tiế
p bằ
ng nhau .
HD :
� =C
� (cu�
� )
Ta co�
:A
ng cha�
n cung BK
1
2
� =C
� (go�
� =C
�
A
c co�
ca�
nh t�
�
ng �
�
ng ) � C
1
1
1
2
� CHK ca�
n �K �
o�
i x�
�
ng v�
�
i H qua BC .
Xe�
t phe�
p�
o�
i x�
�
ng tru�
c BC .
�
�
�
BC H ; B I����
BC B ; C I����
BC C
Ta co�
: K I����
�
BC ��
Va�
y : ��
�
�
ng tro�
n ngoa�
i tie�
p KBC I����
�
�
ng tro�
n ngoa�
i tie�
p HBC
- 15 -
36 Cho ABC va�
�
�
�
�
ng tha�
ng a �
i qua �
�
nh A nh�
ng kho�
ng �
i qua B,C .
a) T�
m a�
nh ABC qua phe�
p�
o�
i x�
�
ng �a.
b) Go�
i G la�
tro�
ng ta�
m ABC , Xa�
c�
�
nh G�
la�
a�
nh cu�
a G qua phe�
p�
o�
i x�
�
ng �a.
Giả
i
a) Vì a làtrục củ
a phé
p đố
i xứ
ng Đa nê
n:
gA �a � A Đa(A) .
gB,C Ͼ�����
a nê
n Đa : B I
b) Vì G Ͼ��
a nê
n Đa :G I
B�
,C I
C�
sao cho a làtrung trực củ
a BB�
,CC�
G�
sao cho a làtrung trực củ
a GG�
.
37 Cho �
�
�
�
ng tha�
ng a va�
hai �
ie�
m A,B na�
m cu�
ng ph�
a�
o�
i v�
�
i a . T�
m tre�
n�
�
�
�
ng
tha�
ng a �
ie�
m M sao cho MA+MB nga�
n nha�
t.
Gia�
i : Xe�
t phe�
p�
o�
i x�
�
ng �a : A I��
� A�
.
M �a th�MA =MA �
. Ta co�
: MA +MB =MA �
+MB �A �
B
�e�
MA +MB nga�
n nha�
t th�cho�
n M,A,B tha�
ng ha�
ng
Va�
y : M la�
giao �
ie�
m cu�
a a va�
A�
B.
38 (SGK-P13)) Cho go�
c nho�
n xOy va�
M la�
mo�
t�
ie�
m be�
n trong go�
c�
o�
. Ha�
y
t�
m�
ie�
m A tre�
n Ox va�
�
ie�
m B tre�
n Oy sao cho MBA co�
chu vi nho�
nha�
t.
Gia�
i
Go�
i N =�Ox(M) va�
P =�Ox(M) . Khi �
o�
: AM=AN , BM=BP
T�
�
�
o�
: CVi =MA+AB+MB =NA+AB+BP �NP
(�
�
�
�
ng ga�
p khu�
c ��
�
�
�
ng tha�
ng )
MinCVi =NP Khi A,B la�
n l�
�
�
t la�
giao �
ie�
m cu�
a NP v�
�
i Ox,Oy .
39 Cho ABC ca�
n ta�
i A v�
�
i�
�
�
�
ng cao AH . Bie�
t A va�
H co�
�
�
nh . T�
m ta�
p h�
�
p
�
ie�
m C trong mo�
i tr�
�
�
ng h�
�
p sau :
a) B di �
o�
ng tre�
n�
�
�
�
ng tha�
ng .
b) B di �
o�
ng tre�
n�
�
�
�
ng tro�
n ta�
m I, ba�
n k�
nh R .
Gia�
i
a) V�
: C =�AH (B) , ma�
B � ne�
n C ��
v�
�
i �
=�AH ( )
Va�
y : Ta�
p h�
�
p ca�
c�
ie�
m C la�
�
�
�
�
ng tha�
ng �
b) T�
�
ng t�
�
: Ta�
p h�
�
p ca�
c�
ie�
m C la�
�
�
�
�
ng tro�
n ta�
m J , ba�
n k�
nh R la�
a�
nh cu�
a
�
�
�
�
ng tro�
n (I) qua �AH .
Vấn đề 4 : PHÉP ĐỐI XỨNG TẤM
A.KIẾN THỨC CƠ BẢN
1 �N : Phe�
p�
o�
i x�
�
ng ta�
m I la�
mo�
t phe�
p d�
�
i h�
nh bie�
n mo�
i�
ie�
m M tha�
nh �
ie�
m M�
�
o�
i x�
�
ng v�
�
i M qua I.
Phe�
p�
o�
i x�
�
ng qua mo�
t�
ie�
m co�
n go�
i la�
phe�
p�
o�
i ta�
m.
�ie�
m I go�
i la�
ta�
m cu�
a cu�
a phe�
p�
o�
i x�
�
ng hay �
�
n gia�
n la�
ta�
m�
o�
i x�
�
ng .
uuur
uuu
r
Kí hiệ
u : ĐI (M) M �
� IM �
IM .
gNe�
u M �I th�M �
�I
gNe�
u M �I th�M �
�I (M) � I la�
trung tr��
c cu�
a MM �
.
g�N :�ie�
m I la�
ta�
m �o�
i x��
ng cu�
a h�
nh H � �I (H) H.
Chu�
y�
: Mo�
t h�
nh co�
the�
kho�
ng co�
ta�
m �o�
i x��
ng .
- 16 -
�I
2 Bie�
u th�
�
c to�
a�
o�
: Cho I(xo;yo) va�
phe�
p�
o�
i x�
�
ng ta�
m I : M(x;y) I���
� M�
�I (M) (x��
;y ) th�
x�
=2xo x
�
��
y 2yo y
�
3 T�
nh cha�
t:
1. Phe�
p�
o�
i x�
�
ng ta�
m ba�
o toa�
n khoa�
ng ca�
ch gi�
�
a hai �
ie�
m ba�
t k�.
2. Bie�
n mo�
t tia tha�
nh tia .
3. Ba�
o toa�
n t�
nh tha�
ng ha�
ng va�
th�
�
t�
�
cu�
a ca�
c�
ie�
m t�
�
ng �
�
ng .
4. Bie�
n mo�
t�
oa�
n tha�
ng tha�
nh �
oa�
n tha�
ng ba�
ng no�
.
5. Bie�
n mo�
t�
�
�
�
ng tha�
ng tha�
nh mo�
t�
�
�
�
ng tha�
ng song song hoa�
c tru�
ng v�
�
i�
�
�
�
ng tha�
ng �
a�
cho .
6. Bie�
n mo�
t go�
c tha�
nh go�
c co�
so�
�
o ba�
ng no�
.
7. Bie�
n tam gia�
c tha�
nh tam gia�
c ba�
ng no�
. ( Tr�
�
c ta�
m � tr�
�
c ta�
m , tro�
ng ta�
m � tro�
ng ta�
m)
8. ��
�
�
ng tro�
n tha�
nh �
�
�
�
ng tro�
n ba�
ng no�
. ( Ta�
m bie�
n tha�
nh ta�
m : I I��
� I�
, R�
=R )
B . BÀI TẬP
1 T�
m a�
nh cu�
a ca�
c�
ie�
m sau qua phe�
p�
o�
i x�
�
ng ta�
mI :
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)
Giả
i:
uur
uur
x�
1 3
x�
4
a) Gỉ
a sử: A �
ĐI (A) � IA IA � (x�
1;y�
2) (3;1) �
�
� A�
(4;1)
y�
2 1
y�
1
Cá
ch �: Dù
ng biể
u thứ
c toạđộ
2 T�
m a�
nh cu�
a ca�
c�
�
�
�
ng tha�
ng sau qua phe�
p�
o�
i x�
�
ng ta�
mI :
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
Gia�
i
PP : Co�
3 ca�
ch
Ca�
ch 1: Du�
ng bie�
u th�
�
c toa�
�
o�
Ca�
ch 2: Xa�
c�
�
nh da�
ng �
// , ro�
i du�
ng co�
ng th�
�
c t�
nh khoa�
ng ca�
ch d(;�
) � �
.
Ca�
ch 3: La�
y ba�
t ky�
A,B � , ro�
i t�
m a�
nh A ��
,B ��
� �
�A �
B�
�I
�
�x 4 x�
x�
4 x
1) Ca�
ch 1: Ta co�
: M(x;y) I���
� M�
��
��
y 2 y �y 2 y�
�
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
�I
Va�
y : ( ) I���
� (�
) : x 2y 5 0
Ca�
ch 2: Go�
i �
=�I () � �
song song � �
: x +2y +m =0 (m �5) .
|5|
| m|
�
m 5 (loa�
i)
Theo �
e�
: d(I; ) =d(I;�
)�
� 5 | m|� �
m 5
�
12 22
12 22
� (�
): x 2y 5 0
Ca�
ch 3: La�
y : A( 5;0),B( 1; 2) � � A �
(9; 2),B�
(5;0) � �
�A ��
B : x 2y 5 0
- 17 -
3 T�
m a�
nh cu�
a ca�
c�
�
�
�
ng tro�
n sau qua phe�
p�
o�
i x�
�
ng ta�
mI :
1) (C) : x2 (y 2)2 1,E(2;1)
2) (C) : x2 y2 4x 2y 0,F(1;0)
3) (P) : y =2x2 x 3 , ta�
m O(0;0) .
� (C�
):(x 4)2 y2 1
� (C�
) : x2 y2 8x 2y 12 0
�/ nghia�
hay bie�
u th�
�
c toa�
�
o�
��������������(P�
): y = 2x2 x 3
HD :1) Co�
2 ca�
ch gia�
i:
Ca�
ch 1: Du�
ng bie�
u th�
�
c toa�
�
o�
.
�E
Ca�
ch 2: T�
m ta�
m I I���
� I�
,R�
R (�
a�
cho) .
2) T�
�
ng t�
�
.
4 Cho hai �
ie�
m A va�
B .Cho bie�
t phe�
p bie�
n�
o�
i M tha�
nh M �
sao cho AMBM �
la�
mo�
t h�
nh b�
nh ha�
nh .
HD :
uuuu
r uuuur
�
�
MA BM �
Ne�
u AMBM �
la�
h�
nh b�
nh ha�
nh � �uuur uuuur
MB AM �
uuuuu
r uuuu
r uuuur uuuu
r uuur �
V�
: MM �
MA AM �
MA MB (1)
uur
uur
Go�
i I la�
trung
�
ie�
m
cu�
a
AB
.
Ta
co�
:
IA
IB
uuuuu
r uuu
r uur uuu
r uur uuuuu
r
uuu
r
�
�
T�
�(1) �uMM
MI
IA
MI
IB
�
MM
2MI
uu
r uuur
� MI IM �
� M�
�I (M) .
5 Cho ba �
�
�
�
ng tro�
n ba�
ng nhau (I1;R),(I 2;R),(I 3;R) t�
�
ng �
o�
i tie�
p
xu�
c nhau ta�
i A,B,C . G�
a s�
�
M la�
mo�
t�
ie�
m tre�
n (I1;R) , ngoa�
i ra :
�I
�C
�A
�B
1� Q .
M I���
� N ; N I���
� P ; P I���
� Q . CMR : M I���
HD :
�Do (I1;R) tie�
p xu�
c v�
�
i (I 2;R) ta�
i A , ne�
n:
uuuur
uuuur
�A
�A
�A
M I�����
N ;I1 I�������
I 2 MI
1 I
NI 2 MI1 NI 2 (1)
�Do (I 2;R) tie�
p xu�
c v�
�
i (I 3;R) ta�
i B , ne�
n:
uuuur
uuur
�B
�B
�B
N I�����
P ;I 2
I �������
I 3 NI
2 I
PI 3 NI 2
PI 3 (2)
�Do (I 3;R) tie�
p xu�
c v�
�
i (I1;R) ta�
i C , ne�
n:
uuur
uuur
�C
�
�C
P I��
�������
Q;I3 I C I1
PI 3 I ��
QI1 � PI 3 QI1 (3)
uuuur
uuur
T�
�
(1),(2),(3) suy ra : MI1 QI1 � M �I (Q) .
1
5 Cho ABC la�
tam gia�
c vuo�
ng ta�
i A . Ke�
�
�
�
�
ng cao AH . Ve�
ph�
a
ngoa�
i tam gia�
c hai h�
nh vuo�
ng ABDE va�
ACFG .
a) Ch�
�
ng minh ta�
p h�
�
p6�
ie�
m B,C,F,G,E,D co�
mo�
t tru�
c�
o�
i x�
�
ng .
b) Go�
i K la�
trung �
ie�
m cu�
a EG . Ch�
�
ng minh K �
�
tre�
n�
�
�
�
ng tha�
ng AH .
c) Go�
i P =DE �FG . Ch�
�
ng minh P �
�
tre�
n�
�
�
�
ng tha�
ng AH .
d) Ch�
�
ng minh : CD BP, BF CP .
e) Ch�
�
ng minh : AH,CD,BF �
o�
ng qui .
- 18 -
HD :
� 45ova�
� 45o ne�
a) Do : BAD
CAF
n ba �
ie�
m D,A,F tha�
ng ha�
ng .
�DF
�DF
�DF
�DF
�Ta co�
: A l����
A ; D l����
D ; F l����
F ; C l����
G;
�DF
B l����
E (T�
nh cha�
t h�
nh vuo�
ng ).
Va�
y : Ta�
p h�
�
p6�
ie�
m B,C,F,G,E,D co�
tru�
c�
o�
i x�
�
ng ch�
nh la�
�
�
�
�
ng tha�
ng DAF .
� AEG.
�
b) Qua phe�
p�
o�
i x�
�
ng tru�
c DAF ta co�
: ABC =AEG ne�
n BAC
� AGE
� ( 2 �
Nh�
ng : BCA
o�
i x�
�
ng =)
� A
� (do KAG ca�
� A
� � K,A,H tha�
AGE
n ta�
i K) . Suy ra : A
ng ha�
ng � K �
�
tre�
n AH .
2
1
2
c) T�
�
gia�
c AFPG la�
mo�
t h�
nh ch�
�
nha�
t ne�
n : A,K,P tha�
ng ha�
ng . (H�
n n�
�
a K la�
trung �
ie�
m cu�
a AP )
Va�
y:P�
�
tre�
n PH .
d) �Do EDC =DBP ne�
n DC =BP .
�
DC =BP
�
� APB
� nh�
�Ta co�
:�
DB =AB � BDC ABP � CD BP � BCD
ng hai go�
c na�
y co�
ca�
p
�
BC =AP
�
ca�
nh : BC AP � ca�
p ca�
nh co�
n la�
i : DC BP.
Ly�
lua�
n t�
�
ng t�
�
, ta co�
: BF CP.
e) Ta co�
: BCP . Ca�
c�
�
�
�
ng tha�
ng AH, CD va�
BF ch�
nh la�
ba �
�
�
�
ng cao cu�
a BCP ne�
n�
o�
ng qui .
6 Cho hai �
ie�
m A va�
B va�
go�
i �A va�
�B la�
n l�
�
�
t la�
hai phe�
p�
o�
i x�
�
ng ta�
m A va�
B.
a) CMR : �B o�A T uuur .
2AB
b) Xa�
c�
�
nh �A o�B.
HD : a) wGo�
i M la�
mo�
t�
ie�
m ba�
t ky�
, ta co�
:
u
u
u
u
r
u
u
u
u
r
�A
M I���
� M�
: MA AM �
uuur uuuuu
r
�B
�
�
�
M�
I���
� M�
: MB BM �
. Ngh�
a la�
: M�
=�B o�A (M),M (1)
�B o�A
�
wTa ch�
�
n
g
minh
:
M
I
�����
�M�
:
uuuuur uuuuu
r uuuuuu
r
�
�
� ��
�
Bie�
t : uMM
uuuu
r MM
uuuu
r MuuM
uuuur
uuuur
�
��
�
�
Ma�: uMM
2MA
va�
M
M
2M
uuuur
uuuu
r uuuur
uuuu
r B uuuur uuur
�
�
�
�
Va�
yu:uu
MM
u2M
u
r uuuu2MA
r
uuu
r Buuu2MA
ur r 2M A 2AB
uuuuur
uuur
�
�
V�
: MA AM �
ne�
n MA M �
A 0 . Suy ra : MM �
2AB � M �
T uuur (M),M (2)
2AB
T�
�
(1) va�
(2) , suy ra : �B o�A T uuur .
2AB
b) Ch�
�
ng minh t�
�
ng t�
�
: �A o�B T uuur .
2BA
7 Ch�
�
ng minh ra�
ng ne�
u h�
nh (H) co�
hai tru�
c�
o�
i x�
�
ng vuo�
ng go�
c v�
�
i nhau th�
(H) co�
ta�
m�
o�
i x�
�
ng .
HD : Du�
ng h�
nh thoi
G�
a s�
�
h�
nh (H) co�
hai tru�
c�
o�
i x�
�
ng vuo�
ng go�
c v�
�
i nhau .
La�
y�
ie�
m M ba�
t ky�
thuo�
c (H) va�
M1 �a(M) , M 2 �b(M1) . Khi �
o�
, theo
�
�
nh ngh�
a M1,M 2 �(H) .
- 19 -
�
�
Go�
i O =a�b , ta co�
: OM =OM1 va�
MOM
1 2AOM1
�OM 2M
� OB
OM1 =OM 2 va�
M
1 2
1
�
�
�
� OB)
Suy ra : OM =OM 2 va�
MOM1 M1OM 2 2(AOM1 +M
1
�
o
o
hay MOM1 2�90 180
Va�
y : O la�
trung �
ie�
m cu�
a M va�
M2 .
Do �
o�
: M 2 �O (M),M �(H),M 2 �(H) � O la�
ta�
m�
o�
i x�
�
ng cu�
a (H) .
� BCN
� =30o th�ABC �e�
8 Cho ABC co�
AM va�
CN la�
ca�
c trung tuye�
n . CMR : Ne�
u BAM
u.
HD :
�
� 30o ne�
�
�
T�
�
gia�
c ACMN co�
NAM
NCM
n no�
i tie�
p�
tro�
n ta�
m O, bk�
nh R=AC va�
MON
2NAM
60o.
�N
�N
Xe�
t : A I���
� B �����
(O) I
(O1) th�B (O1) v�A (O) .
�M
�
C I������
B (O)
���
I M
(O2) th�B (O2) v�C (O) .
�
OO OO2 2R
�
Khi �
o�
, ta co�
:� 1
� OO1O2 la�
tam gia�
c�
e�
u.
�
o
�MON 60
V�O1B O2B R R 2R O1O2 ne�
n B la�
trung �
ie�
m O1O2.
Suy ra :ABC ; OO1O2 (V�cu�
ng �
o�
ng da�
ng v�
�
i BMN) .
V�OO1O2 la�
tam gia�
c�
e�
u ne�
n ABC la�
tam gia�
c�
e�
u.
- 20 -
Vấn đề 5 : PHÉP QUAY
A. KIẾN THỨC CƠ BẢN
1 �N : Trong ma�
t pha�
ng cho mo�
t �ie�
m O co�
��
nh va�
go�
c l�
��
ng gia�
c . Phe�
p bie�
n h�
nh bie�
n mo�
i �ie�
m
M tha�
nh �
ie�
m M �sao cho OM =OM�
va�
(OM;OM �
) = ��
��
c go�
i la�
phe�
p quay ta�
m O v��
i go�
c quay .
gPhe�
p quay hoa�
n toa�
n xa�
c ��
nh khi bie�
t ta�
m va�
go�
c quay
gK�hie�
u : Q
O .
Chu�
y�
: Chie�
u d��ng cu�
a phe�
p quay �chie�
u d��ng cu�
a ����
ng tro�
n l��ng gia�
c.
gQ2k �phe�
p �o�
ng nha�
t ,k ��
gQ(2k+1) �phe�
p �o�
i x��
ng ta�
m I ,k ��
2 T�
nh cha�
t:
g�L : Phe�
p quay la�
mo�
t phe�
p d��
i h�
nh .
gHQ :
1.Phe�
p quay bie�
n ba �ie�
m tha�
ng ha�
ng tha�
nh ba �ie�
m tha�
ng ha�
ng va�
ba�
o toa�
n th��
t��
cu�
a ca�
c �ie�
m t��ng
��
ng .
2. ����
ng tha�
ng tha�
nh ����
ng tha�
ng .
3. Tia tha�
nh tia .
4. �oa�
n tha�
ng tha�
nh �oa�
n tha�
ng ba�
ng no�
.
Q
Q
5. Tam gia�
c tha�
nh tam gia�
c ba�
ng no�
. (Tr��
c ta�
mI���
� tr�
�
c ta�
m , tro�
ng ta�
mI���
� tro�
ng ta�
m)
Q(O ; )
6. ��
��
ng tro�
n tha�
nh �
�
��
ng tro�
n ba�
ng no�
. ( Ta�
m bie�
n tha�
nh ta�
m : I I�����
�I�
, R�
=R )
7. Go�
c tha�
nh go�
c ba�
ng no�
.
B. BÀI TẬP
1 Trong ma�
t pha�
ng Oxy cho �
ie�
m M(x;y) . T�
m M / =Q(O ; )(M) .
HD :
�
x =rcos
Go�
i M(x;y) . �a�
t : OM =r , go�
c l�
�
�
ng gia�
c (Ox;OM) = th�
M�
y =rsin
�
Q(O ; )
V�: M I�����
� M / . Go�
i M / (x��
;y ) th��
o�
da�
i OM / =r va�
(Ox;OM / ) = + .
Ta co�
:
x�
=rcos( +) =acos.cos asin.sin xcos ysin .
y�
=rsin( +) =asin.cos acos.sin xsin ycos .
�
x�
=xcos ysin
Va�
y : M/ �
y�
=xsin ycos
�
�a�
c bie�
t:
Q(O ; )
�
�
x�
=xcos ysin
w M I�����
� M // �
�
y�
= xsin ycos
�
Q(I ; )
x�
xo =(x xo)cos (y yo)sin
�
wM I�����
�M/ �
y�
yo =(x xo)sin (y yo)cos
I(xo;yo)
�
Q(I ; )
�
�
x�
xo =(x xo)cos (y yo)sin
wM I�����
� M // �
�
y�
yo = (x xo)sin (y yo)cos
I(xo;yo)
�
- 21 -
2 Trong mpOxy cho phe�
p quay Q
. T�
m a�
nh cu�
a:
(O;45o)
a) �ie�
m M(2;2)
b) ��
�
�
ng tro�
n (C) : (x 1)2+y2=4
Q
(O; 45o)
Gia�
i . Go�
i : M(x;y) I������ M / (x/ ;y/ ) . Ta co�
: OM =2 2, (Ox; OM) =
�
�
x�
=rcos(+45o) rcos.cos45o rsin.sin45o x.cos45o y.sin45o
Th�M / �
�
y�
=rsin(+45o) rsin.cos45o rcos.sin45o y.cos45o x.sin45o
�
�
2
2
x�
=
x
y
�
�
2
2
� M/ �
2
2
�
y�
=
x
y
�
�
2
2
Q
(O ; 45o)
a) A(2;2) I������ A / (0 ;2 2)
Q
�
�
�
(O; 45o)
gTa�
m I(1;0)
gTa�
m I /?
b) V�(C) : �
������ (C�
): �
gBk : R =2
gBk : R�
=R =2
�
�
Q
2
2
22
22
(O ; 45o)
I(1;0)I������ I / (
;
) . Va�
y : (C�
) : (x
) +(y
) =4
2
2
2
2
� 1
3
x�
= x
y
�
�
2 . Ho�
3 Trong mpOxy cho phe�
p bie�
n h�
nh f : � 2
i f la�
phe�
p g�?
3
1
�
y�
=
x y
�
�
2
2
Gia�
i
�
x�
=xcos ysin
�
�
3
3 � f la�
Ta co�
f : M (x;y) I��
� M ���
(x ;y ) v�
�
i �
phe�
p quay Q
(O; )
�
y�
=xsin ycos
3
�
3
3
4 Trong mpOxy cho đườ
ng thẳ
ng () : 2x y+1=0 . Tìm ả
nh củ
a đườ
ng thẳ
ng qua :
a) Phé
p đố
i xứ
ng tâ
m I(1; 2).
b) Phé
p quay Q
.
(O;90o)
Giả
i
�
�
x�
2 x
x 2 x�
a) Ta có: M ���
(x ;y ) =ĐI (M) thì biể
u thứ
c tọa độM �
��
�
�
y
4
y
y 4 y�
�
�
Vì M(x;y) �() : 2x y+1=0 � 2(2 x�
) (4 y�
) 1 0 � 2x�
y�
9 0
� M ���
(x ;y ) �(�
): 2x y 9 0
ĐI
Vậ
y : ( ) I���
� (�
) : 2x y 9 0
Q
(O;90o)
b) Cá
ch 1 : Gọi M(x;y) I�����
� M ���
(x ;y ) . Đặ
t (Ox ; OM) = , OM =r ,
Ta có(Ox ; OM �
) = + 90o,OM �
r.
Q
�
�
�
x =rcos
x�
rcos( 90o) rsin y �
x y�
(O;90o)
Khi đó: M �
I�����
� M�
��
�
y
=
rsin
o
y x�
�
�
y�
rsin( 90 ) rcos x
�
�
Vì M(x;y) �( ) : 2(y�
) ( x�
) +1 =0 � x�
2y�
+1 =0 � M ���
(x ;y ) �(�
): x 2y 1 0
Q
(O;90o)
Vậ
y : () I�����
� (�
): x 2y 1 0
- 22 -
Q
(O;90o)
Ca�
ch 2 : La�
y : �M(0;1) �() I�����
� M�
(1;0) �(�
)
Q
1
1
(O;90o)
�N( ;0) �() I�����
� N�
(0; ) �(�
)
2
2
Q
(O;90o)
�() I�����
� (�
) �M ��
N : x 2y 1 0
Q
1
(O;90o)
Ca�
ch 3 : �V�
() I�����
�(�
) � () (�
) ma�
he�
so�
go�
c : k 2 � k�
2
Q
(O;90o)
�M(0;1) �() I�����
� M�
(1;0) �(�
)
�
gQua M�
(1;0)
�
�(�
): �
) : x 2y 1 0
1 � (�
g
hsg
;
k
=
�
�
2
5 Trong ma�
t pha�
ng toa�
�
o�
Oxy cho A(3;4) . Ha�
y t�
m toa�
�
o�
�
ie�
m A�
la�
a�
nh
o
cu�
a A qua phe�
p quay ta�
m O go�
c 90 .
HD :
Go�
i B(3;0),C(0;4) la�
n l�
�
�
t la�
h�
nh chie�
u cu�
a A le�
n ca�
c tru�
c Ox,Oy . Phe�
p
quay ta�
m O go�
c 90o bie�
n h�
nh ch�
�
nha�
t OABC tha�
nh h�
nh ch�
�
nha�
t OC���
A B.
Khi �
o�
: C�
(0;3),B�
( 4;0). Suy ra : A �
( 4;3).
6 Trong ma�
t pha�
ng toa�
�
o�
Oxy . T�
m phe�
p quay Q bie�
n�
ie�
m A( 1;5)
tha�
nh �
ie�
m B(5;1) .
uuur
uuur
�
�
OA OB 26
HD : Ta co�
: OA (1;5) va�
OB (5;1) � �uuur uuur
OA.OB 0 � OA OB
�
� B =Q
(A) .
(O ; 90o)
7 Trong ma�
t pha�
ng toa�
�
o�
Oxy , cho �
ie�
m M(4;1) . T�
m N =Q
(M) .
(O ; 90o)
HD :
uuuu
r uuur
o � OM.ON =0 � 4x+y =0 � y= 4x (1)
V�N =Q
(M)
�
(OM;ON)
90
(O ; 90o)
Do : OM ON � x2 y2 16 1 17 (2) .
Gia�
i (1) va�
(2) , ta co�
: N(1; 4) hay N( 1;4) .
wTh�
�
la�
i : �ie�
u kie�
n (OM;ON) 90o ta tha�
y N( 1;4) thoa�
ma�
n.
8 a)Trong ma�
t pha�
ng toa�
�
o�
Oxy , cho �
ie�
m A(0;3) . T�
m B =Q
(A) .
(O ; 45o)
HD : Phe�
p quay Q
bie�
n�
ie�
m A �Oy tha�
nh �
ie�
m B ��
t: y x,ta co�
:
(O ; 45o)
�
x yB 0
3
3 3
. Ma�OB = x2
y2
3 � xB
� B( ; ).
�B
B
B
OA OB 3
2
2 2
�
b) Cho A(4;3) . T�
m B =Q
(A)
(O;60o)
��
�B (
4 3 3 3 4 3
;
)
2
2
- 23 -
9 Cho �
�
�
�
ng tro�
n (C) : (x 3)2 (y 2)2 4 . T�
m (C�
) =Q
(C) .
(O ; 90o)
HD : T�
m a�
nh cu�
a ta�
mI : Q
(I) I �
(2;3) � (C�
):(x 2)2 (y 3)2 4 .
(O ; 90o)
10 Cho �
�
�
�
ng tro�
n (C) : (x 2)2 (y 2 3)2 5 . T�
m (C�
) =Q
(C) .
(O ; 60o)
HD : T�
m a�
nh cu�
a ta�
mI : Q
(I) I �
(2;2 3) � (C�
):(x 2)2 (y 2 3)2 5 .
o
(O ; 60 )
11 Cho �
�
�
�
ng tro�
n (C) : (x 2)2 (y 2)2 3 . T�
m (C�
) =Q
(C) .
(O ; 45o)
HD : T�
m a�
nh cu�
a ta�
mI : Q
(I) I �
(1 2;1 2) � (C�
):(x 1 2)2 (y 1 2)2 3 .
(O ; 45o)
12 [CB-P19] Trong ma�
t pha�
ng toa�
�
o�
Oxy , cho �
ie�
m A(2;0) va�
�
�
�
�
ng tha�
ng (d) : x +y 2 =0.
T�
m a�
nh cu�
a A va�
(d) qua phe�
p quay Q
.
(O ; 90o)
HD :
wTa co�
: A(2;0) �Ox . Go�
i B =Q
(A) th�
B �Oy va�
OA =OB .
(O ; 90o)
wV�toa�
�
o�
A,B thoa�
ma�
n pt (d) : x +y 2 =0 ne�
n A,B �(d) .
Do B =Q
(A) va�
t�
�
ng t�
�
Q
(A) =C( 2;0)
(O ; 90o)
(O ; 90o)
x
y
x y
ne�
nQ
(d) =BC � (BC) :
1�
1� x y 2 0
o
(O ; 90 )
xC yC
2 2
13 Cho (d) : x 3y 1 =0 . T�
m =Q
(d) .
� ( ) : 3x y 1 0
(O ; 90o)
14 Cho (d) : 2x y 2 =0 . T�
m =Q
(d) .
(O ; 60o)
1 3
a�
nh
HD : d �Ǿ���
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 gia�
c�
e�
u ABC co�
ta�
m O va�
phe�
p quay Q
.
(O;120o)
a) Xa�
c�
�
nh a�
nh cu�
a ca�
c�
�
nh A,B,C .
b) T�
m a�
nh cu�
a ABC qua phe�
p quay Q
(O;120o)
Gia�
i
� BOC
� COA
� 120one�
a) V�
OA =OB =OC va�
AOC
nQ
: A I��
� B,B I��
� C,C I��
�A
(O;120o)
b) Q
: ABC ��
� ABC
(O;120o)
16 [CB-P19] Cho hình vuô
ng ABCD tâ
mO .
a) Tìm ả
nh củ
a điể
m C qua phé
p quay Q
.
(A ; 90o)
b) Tìm ả
nh củ
a đườ
ng thẳ
ng BC qua phé
p quay Q
(O ; 90o)
�
o n AEC
HD : a) Gọi E =Q
o (C) thì AE=AC vàCAE 90 nê
(A ; 90 )
vuô
ng câ
n đỉ
nh A , cóđườ
ng cao AD . Do đó: D làtrung điể
m củ
a EC .
b) Ta có: Q
(B) C vàQ
(B) C � Q
(BC) CD.
(O ; 90o)
(O ; 90o)
(A ; 90o)
- 24 -
17 Cho h�
nh vuo�
ng ABCD ta�
m O . M la�
trung �
ie�
m cu�
a AB , N la�
trung �
ie�
m
cu�
a OA . T�
m a�
nh cu�
a AMN qua phe�
p quay Q
.
(O;90o)
HD : w Q
(A) D , Q
(M) M �
la�
trung �
ie�
m cu�
a AD .
(O;90o)
(O;90o)
Q
(N) N�
la�
trung �
ie�
m cu�
a OD . Do �
o�
:Q
(AMN) DM ��
N
(O;90o)
(O;90o)
18 [ CB-1.15 ] Cho h�
nh lu�
c gia�
c�
e�
u ABCDEF , O la�
ta�
m�
��
�
ng tro�
n ngoa�
i tie�
p cu�
a no�
. T�
m a�
nh cu�
a
OAB qua phe�
p d�
�
i h�
nh co�
�
�
�
�
c ba�
ng ca�
ch th�
�
c hie�
n lie�
n tie�
p phe�
p quay ta�
m O , go�
c 60o va�
phe�
p
uuur .
t�
nh tie�
n TOE
HD :
uuur oQ
Go�
i F =TOE
. Xe�
t:
(O;60o)
(O) O,Q
(A) B,Q
(B) C .
(O;60o)
(O;60o)
(O;60o)
uuur (O) E,Tuuur (B) O,Tuuur (C) D
wTOE
OE
OE
wQ
wVa�
y : F(O) =E , F(A) =O , F(B) =D � F(OAB) =EOD
19 Cho h�
nh lu�
c gia�
c�
e�
u ABCDEF theo chie�
u d�
�
ng , O la�
ta�
m�
�
�
�
ng tro�
n ngoa�
i tie�
p cu�
a no�
. I la�
trung �
ie�
m cu�
a AB .
a) T�
m a�
nh cu�
a AIF qua phe�
p quay Q
.
(O ; 120o)
b) T�
m a�
nh cu�
a AOF qua phe�
p quay Q
.
(E ; 60o)
HD :
a) wQ
bie�
n F,A,B la�
n l�
�
�
t tha�
nh B,C,D , trung �
ie�
mI
(O ; 120o)
tha�
nh trung �
ie�
m J cu�
a CD ne�
nQ
(AIF) CJ B .
(O ; 120o)
b) wQ
bie�
n A,O,F la�
n l�
�
�
t tha�
nh C,D,O .
(E ; 60o)
15 Cho ba điể
m A,B,C theo thứtựtrê
n thẳ
ng hà
ng . Vẽcù
ng mộ
t phía dựng hai tam giá
c đề
u ABE và
BCF . Gọi M vàN tương ứ
ng làhai trung điể
m củ
a AF vàCE . Chứ
ng minh rằ
ng : BMN làtam giá
c đề
u.
HD :
Xé
t phé
p quay Q
.Ta có: Q
(A) E , Q
(F) C
(B;60o)
(B;60o)
(B;60o)
�Q
(AF) EC .
(B;60o)
Do M làtrung điể
m củ
a AF , N làtrung điể
m củ
a EC , nê
n:
o
� 60 � BMN làtam giá
Q
(M) N � BM =BN vàMBN
c đề
u.
(B;60o)
21 [ CB-1.17 ] Cho n�
�
a�
�
�
�
ng tro�
n ta�
mO �
�
�
�
ng k�
nh BC . �ie�
m A cha�
y tre�
n n�
�
a�
�
�
�
ng tro�
n�
o�
.
D�
�
ng ve�
ph�
a ngoa�
i cu�
a ABC h�
nh vuo�
ng ABEF . Ch�
�
ng minh ra�
ng : E cha�
y
tre�
n n�
�
a�
�
�
�
ng co�
�
�
nh .
HD : Go�
i E =Q
(A)
. Khi A cha�
y tre�
n n�
�
a�
�
�
�
ng tro�
n (O) ,
(B;90o)
E se�
cha�
y tre�
n n�
�
a�
�
�
�
ng tro�
n (O�
) =Q
[(O)] .
(B;90o)
22 Cho �
�
�
�
ng (O;R) va�
�
�
�
�
ng tha�
ng kho�
ng ca�
t�
�
�
�
ng tro�
n . Ha�
y
d�
�
ng a�
nh cu�
a () qua phe�
p quay Q
.
(O ; 30o)
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