Dilations
• Determine whether a dilation is an enlargement, a reduction,
or a congruence transformation.
• Determine the scale factor for a given dilation.
Vincent Van Gogh’s “Starry
Night” reduced by a factor of 2
CLASSIFY DILATIONS
A dilation is a transformation that may change the size of a
figure.
A dilation requires a center point and a scale factor.
CLASSIFY DILATIONS
A dilation is a transformation that may change the size of a
figure.
A dilation requires a center point and a scale factor.
Example:
A’
CA’ = 2(CA)
CB’ = 2(CB)
CD’ = 2(CD)
B’
r=2
A
B
center
C
D
D’
CLASSIFY DILATIONS
A dilation is a transformation that may change the size of a
figure.
A dilation requires a center point and a scale factor.
Another example:
XM’ = ⅓(XM)
r = 1/3
M
N
XN’ = ⅓(XN)
XO’ = ⅓(XO)
XP’ = ⅓(XP)
P
M’
P’
O
N’
O’
X
center
Key Concept
Dilation
If |r| > 1, the dilation is an enlargement.
If 0 < |r| < 1, the dilation is a reduction.
If |r| = 1, the dilation is a congruence transformation.
Key Concept
Dilation
We have seen that dilations produce similar figures,
therefore dilation is a similarity transformation.
A’
AB BD AD


AB
BD
AD
B’
A
C
B
D
D’
and
M
M N  N O OP M P



MN
NO
OP
MP
P
N
M’
P’
O
N’
O’
Theorem
If a dilation with center C and a scale factor of r
transforms A to E and B to D, then ED = |r|(AB)
A
D
B
E
C
Example 1
Determine Measures
Under Dilations
Find the measure of the dilation image A’B’ or the preimage
AB using the given scale factor.
a. AB = 12, r = -2
b. A’B’ = 36, r = ¼
A’B’ = |r|(AB)
A’B’ = |r|(AB)
A’B’ = 2(12)
36 = ¼(AB)
A’B’ = 24
AB = 144
When the scale factor is negative, the image falls on
the opposite side of the center than the preimage.
Key Concept
Dilations
If r > 0, P’ lies on CP, and CP’ = r ∙ CP
If r < 0, P’ lies on CP’ the ray opposite CP, and CP’ = r ∙ CP
The center of dilation is always its own image.
Example 2
Draw a Dilation
Draw the dilation image of ∆JKL with center C and r = ½
K
J
L
Example 2
Draw a Dilation
Draw the dilation image of ∆JKL with center C and r = -½
K
Since 0 < |r| < 1, the
dilation is a reduction
J
L
K
C
J
L
L’
J’
K’
Theorem
If P(x, y) is the preimage of a dilation centered at the origin
with scale factor r, then the image is P’(rx, ry).
Example 3 Dilations in the Coordinate Plane
COORDINATE GEOMETRY Triangle ABC has vertices
A(7, 10), B(4, -6), and C(-2, 3). Find the image of ∆ABC after
a dilation centered at the origin with the scale factor of 2.
Sketch the preimage and the image.
22
20
18
16
14
12
10
8
6
4
2
-10
-5
-2
-4
-6
-8
-10
-12
5
10
15
Example 3 Dilations in the Coordinate Plane
COORDINATE GEOMETRY Triangle ABC has vertices
A(7, 10), B(4, -6), and C(-2, 3). Find the image of ∆ABC after
a dilation centered at the origin with the scale factor of 2.
Sketch the preimage and the image.
Preimage
(x, y)
Image
(2x, 2y)
A(7, 10)
A’(14, 20)
B(-4, 6)
B’(8, -12)
C(-2, 3)
C’(-4, 6)
22
20
18
16
14
12
10
8
6
4
2
-10
-5
-2
-4
-6
-8
-10
-12
5
10
15
Example 4
Identify the Scale Factor
Determine the scale factor for each dilation with center C.
Then determine whether the dilation is an enlargement,
reduction, or congruence transformation.
a.
9
A’
B’
8
Scale factor =
image length
preimage length
B
A
7
6
C
5
= 6 units
3 units
4
E
3
2
=2
D
D’
E’
1
2
4
6
8
10
12
Example 4
Identify the Scale Factor
Determine the scale factor for each dilation with center C.
Then determine whether the dilation is an enlargement,
reduction, or congruence transformation.
a.
9
8
Scale factor =
image length
preimage length
H
G
7
6
C
5
= 4 units
4 units
4
J
3
F
2
=1
1
2
4
The dilation is a congruence transformation
6
8
10
12