Lesson 29
TAKS Grade 9 Objective 6
(8.6)(A)
Generating Similar Figures
Using Dilations
A dilation is a transformation that enlarges or reduces a figure to make a
similar image. An enlargement makes a new image that is larger than
the original figure. A reduction makes a new image that is smaller than the
original figure. The ratio of the dimensions of the new image to those of
the original figure is called the scale factor.
Finding Dilation Images
New Vocabulary
•
•
•
•
dilation
enlargement
reduction
scale factor
To find the dimensions of the dilated image, multiply the lengths of the sides
of the original figure by the scale factor.
EXAMPLE 1
Quadrilateral JKLM is dilated to form quadrilateral PQRS using a scale factor of 2.
Find the lengths of the sides of quadrilateral PQRS.
K 1
L
3
4
J
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2
M
Multiply the length of each side of quadrilateral JKLM by the scale factor 2 to obtain
the lengths of the sides of quadrilateral PQRS.
PQ 2 JK 2 3 6
QR 2 KL 2 1 2
RS 2 LM 2 2 4
PS 2 JM 2 4 8
Quadrilateral PQRS is shown below.
Q
2
6
To find the coordinates
of an image after
dilation, multiply the
coordinates of the
original figure by the
scale factor.
R
4
8
P
S
Quick Check 1
1a. The side lengths of quadrilateral ABCD are
4, 6, 8, and 10 units. Find the side lengths for
the dilation of quadrilateral ABCD with a
scale factor of 34.
TAKS Review and Preparation Workbook
1b. The lengths of the sides of DEF are 3, 4,
and 5 units. Find the side lengths for the
dilation of DEF with a scale factor of 1.5.
LESSON 29
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Generating Similar Figures Using Dilations
85
TAKS Objective 6 (8.6)(A)
LESSON 29
Finding the Scale Factor of a Dilation
To find the scale factor of a dilation, divide a length of the dilated image by
the corresponding length of the original image. If the scale factor is less than
1, the dilation is a reduction. If the scale factor is greater than 1, the dilation
is an enlargement.
EXAMPLE 2
XYZ is dilated to ABC. Find the scale factor of the dilation and
determine whether the dilation is an enlargement or a reduction.
An enlargement is a
dilation in which the
scale factor is greater
than 1. A reduction is
a dilation in which the
scale factor is less
than 1.
Y
5
4
6
X
B
2 12
A
Z
2
3
C
Quick Check 2
2a. Determine the scale factor for the dilation
of the equilateral triangle LMN to PQR.
Q
M
4
6
L
P
86
LESSON 29
■
2b. Rectangle ABCD is 6 cm wide and 2 cm high.
The rectangle is dilated to form rectangle
EFGH, which is 9 cm wide and 3 cm high.
Determine the scale factor for the dilation,
and state whether it is an enlargement or a
reduction.
N
R
Generating Similar Figures Using Dilations
TAKS Review and Preparation Workbook
Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved.
Step 1 To find the scale factor of the dilation, divide the length of AC by the
length of XZ.
AC 3 1
XZ 6 2
Step 2 Because the scale factor of the dilation is less than 1, this dilation is a reduction.
Name__________________________Class____________Date________
1 Rectangle ABCD is 3 inches wide and
6 inches high. The rectangle is reduced to
rectangle EFGH, which is 1 inch wide and
2 inches high. What is the scale factor for
the reduction?
4 Pentagon JKLMN is dilated using a scale
factor of 2 to form pentagon PQRST. Which
of the following best represents pentagon
PQRST?
L
A 13
K
B 12
M
C 2
J
N
1.0
D 3
F
R
Q
2 Find the coordinates for the dilation of
RST by a scale factor of 2. The coordinates
of RST are R(1.5, 1), S(2, 21), and
T (1, 12).
S
P 0.5 T
G
F R(2, 2), S(4, 2), T(2, 2)
R
Q
G R(1, 0.5), S(1.5, 2), T(0.5, 1)
S
Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved.
H R(2, 0.5), S(2, 1), T(0.5, 2)
P
J R(3, 2), S(4, 1), T(2, 1)
1.2
H
3 GHJ is dilated to form KLM.
4
R
S
Q
y
H
G
P
L
K
–4
T
2
O
4x
J
2.0
T
R
M
–4
Q
J
S
What is the scale factor for the dilation?
A 1
C 2
B 12
D 3
TAKS Review and Preparation Workbook
P
LESSON 29
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3.0
T
Generating Similar Figures Using Dilations
87