GEOMETRY
WORKSHEET – Dilations
Jan. 18, 2013
A dilation, 𝐷𝑂,𝑘 with center 𝑂 and scale factor 𝑘 maps any point 𝑃 to a point 𝑃′ as determined by
(1) if 𝑘 > 0, 𝑃′ lies on ⃗⃗⃗⃗⃗
𝑂𝑃 and 𝑂𝑃′ = 𝑘 ∙ 𝑂𝑃
B'
C'
B
⃗⃗⃗⃗⃗ and 𝑂𝑃′ = |𝑘| ∙ 𝑂𝑃
(2) if 𝑘 < 0, 𝑃′ lies on the ray opposite 𝑂𝑃
C
(3) the center 𝑂 is its own image.
𝐷𝑂,2 : ∆𝐴𝐵𝐶 → ∆𝐴′𝐵′𝐶′
M
If |𝑘| < 1 the dilation is called a contraction.
A'
A
O
If |𝑘| > 1 the dilation is called an expansion.
L
N
P'
K'
X
The examples at right show some typical dilations.
L'
N'
K
P
M'
𝐷𝑋,−1 : 𝐾𝐿𝑀𝑁𝑃 → 𝐾′𝐿′𝑀′𝑁′𝑃′
2
Obviously, a dilation is not an isometry (except in the special case where |𝑘| = 1). Since a dilation
maps a figure to a similar figure, a dilation is often called a similarity mapping.
Theorem: A dilation maps any triangle to a similar triangle.
Corollary: A dilation maps an angle to a congruent angle.
Corollary: A dilation 𝐷𝑃,𝑘 maps any segment to a parallel segment |𝑘| times as long.
Corollary: A dilation 𝐷𝑃,𝑘 maps any figure of area 𝐴 to a similar figure whose area is 𝐴 ∙ 𝑘 2 .
Exercises:
1. Find the coordinates of the images of 𝐴, 𝐵, and 𝐶 by the given dilation.
8
a. 𝐷𝑂,2
b. 𝐷𝑂,3
6
4
c. 𝐷𝑂,1
2
d. 𝐷𝑂,−1
2
-5
e. 𝐷𝐴,2
f. 𝐷𝐴,−1
O
-2
-4
g. 𝐷𝐵,1
h. 𝐷𝐶,1
B (4, 2)
2
-6
2
-8
5
C (2, -2)
A (6, 0)
10
2. The mappings below represent dilations centered at the origin. Find the scale factor of the dilation
and state whether it is an expansion or a contraction.
a. (2, 0) → (8, 0)
b. (2, 3) → (4, 6)
c. (3, 9) → (1, 3)
d. (4, 10) → (−2, −5)
1
2
f. (−6, 2) → (18, −6)
e. (0, 6) → (0, 3)
3. Graph quad. 𝑃𝑄𝑅𝑆 and its image under the given dilation. Then find the ratio of the perimeters
and the ratio of the areas of the two quadrilaterals.
a. 𝑃(−1, 1); 𝑄(0, −1); 𝑅(4, 0); 𝑆(2, 2) 𝐷𝑂,3
b. 𝑃(12, 0); 𝑄(0, 15); 𝑅(−9, 6); 𝑆(3, −9)
𝐷𝑂,2
3
c. 𝑃(3, 0); 𝑄(3, 4); 𝑅(6, 6); 𝑆(5, −1)
𝐷𝑂,−2
d. 𝑃(−2, −2); 𝑄(0, 0); 𝑅(4, 0); 𝑆(6, −2)
𝐷𝑂,−1
2
4. Graph the image of each given triangle under the stated dilation:
a. 𝐷𝐴,2
b. 𝐷𝑊,−1
-25
-20
c. 𝐷𝑃,−1
-15
2
-10
-5
5
-2
16
-4
14
-6
X
B
W
12
-8
P
Y
10
H
G
-10
8
C
A
-12
F
Z6
-14
4
5. State another name for 𝐷𝑃,−1.
2
5
10
15
20
O
25
5 30
10
15
20