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9.1 – Translate Figures and Use Vectors
Transformation: Moves or changes a figure
Preimage: Original figure
Image: Transformed figure
Point P becomes P '
“P prime”
Isometry: A congruent transformation
Translation:
An isometry that moves every point a
certain distance in a certain direction
P'
P
Q'
Q
Translation:
Note:
PP ' QQ ' and PP '  QQ '
Motion Rule:
Moves each point left, right, down, or up
( x, y)   x  #, y  #
Left
or
Right
Down
or
Up
Use the translation ( x, y)   x  2, y  5
What is the image of D(4, 7)?
D '  (4 + 2, 7 – 5)
D +2
–5
D'
D '  (6, 2)
Use the translation ( x, y)   x  7, y  4
What is the image of R(2, –4)?
R '  (2 – 7, –4 + 4)
R '  (–5, 0)
R'
+4
–7
R
Use the translation ( x, y)   x  4, y  6
What is the preimage of M ' (–5, 3)?
M
M  (–5 – 4, 3 + 6)
+6
M  (–9, 9)
–4
M'
Use the translation ( x, y)   x  5, y 
What is the preimage of A '(4, –1)?
A  (4 + 5, –1)
A  (9, –1)
A'
+5 A
The vertices of ABC are A(–1, 1), B(4, –1), and C(2, 4).
Graph the image of the triangle using prime notation.
( x, y )   x  3, y  5
C'
A'
A '  (–1 – 3, 1 + 5)
A '  (–4, 6)
B' C
B '  (4 – 3, –1 + 5)
A
B '  (1, 4)
B
C '  (2 – 3, 4 + 5)
C '  (–1, 9)
The vertices of ABC are A(–1, 1), B(4, –1), and C(2, 4).
Graph the image of the triangle using prime notation.
( x, y )   x, y  3 
A '  (–1 , 1 – 3)
A '  (–1, –2)
A
A'
C
C'
B '  (4, –1 – 3)
B
B'
B '  (4, –4)
C '  (2, 4 – 3)
C '  (2, 1)
 A ' B ' C ' is the image of ABC after a
translation. Write a rule for the translation.
( x, y )   x  5, y  3
+3
–5
 A ' B ' C ' is the image of ABC after a
translation. Write a rule for the translation.
( x, y )   x  2, y  5
+2
–5
Vector:
Translates a shape in direction and magnitude,
or size.
Written: FG
Where F is the
initial point and
G is the terminal
point.
Vector:
Component form: < x, y >
5, 3
Name the vector and write its component form.
JD
5, 1
+5
–1
Name the vector and write its component form.
7,  3
DR
–7
–3
Name the vector and write its component form.
RS
0,  4
–4
Use the point P(5, –2). Find the component form
of the vector that describes the translation to P '.
P '(2, 0)
3, 2
P'
+2
–3 P
Use the point P(5, –2). Find the component form
of the vector that describes the translation to P '.
P '(5,  4)
10,  2
–10
–2
P'
P
Find the value of each variable in the translation.
a = 80°
2b = 8
b=4
c = 13
5d = 100
d = 20°
Find the value of each variable in the translation.
b – 5 = 12
b = 17
a = 180 – 90 – 31
a = 59°
3c + 2 = 20
3c = 18
c=6
9.3 – Perform Reflections
Reflection:
Transformation that uses a line like a mirror
to reflect an image
Line of Reflection:
Mirror line in a reflection
A reflection in a line m maps every point P in
the plane to a point P ' , such that:
• If P is not on m, then m is the perpendicular
bisector of PP '.
• If P is on m, then P  P '
Reflect point P(5, 7) in the given line.
x – axis
P(5, 7) becomes
P '(5,  7)
P
P'
A reflection in the
x-axis changes
(x, –y)
(x, y) into _______
Reflect point P(5, 7) in the given line.
y – axis
P(5, 7) becomes
P '( 5, 7)
P'
P
A reflection in the
y-axis changes
(–x, y)
(x, y) into _______
Reflect point P(5, 7) in the given line.
y=x
P(5, 7) becomes
P '(7, 5)
P
P'
A reflection in the
y = x changes
(y, x)
(x, y) into _______
Graph the reflection of the polygon in the given line.
x – axis
C'
A'
B'
Graph the reflection of the polygon in the given line.
y – axis
D'
C'
A'
B'
Graph the reflection of the polygon in the given line.
y=x
A  (–1 , –3)
C'
B'
A '  (–3, –1)
B  (2, –4 )
A'
B '  (–4, 2)
C  (3, 0)
C '  (0, 3)
Graph the reflection of the polygon in the given line.
x – axis
B'
C'
A'
D'
Graph the reflection of the polygon in the given line.
x = –1
A'
C'
B'
Graph the reflection of the polygon in the given line.
y=2
B'
A' 
C'
 D'