NAME
DATE
7-6
PERIOD
Study Guide and Intervention
Identify Similarity Transformations A dilation is a transformation that enlarges
or reduces the original figure proportionally. The scale factor of a dilation, k, is the ratio
of a length on the image to a corresponding length on the preimage. A dilation with k > 1 is
an enlargement. A dilation with 0 < k < 1 is a reduction.
Example
Determine whether the dilation from A to B is an enlargement or a
reduction. Then find the scale factor of the dilation.
a.
b.
y
y
"
#
"
x
0
x
0
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
#
B is larger than A, so the dilation is
an enlargement.
B is smaller than A, so the dilation is a
reduction.
The distance between the vertices at
(-3, 4) and (-1, 4) for A is 2. The
distance between the vertices at
(0, 3) and (4, 3) for B is 4.
The distance between the vertices at (2, 3) and
(2, -3) for A is 6. The distance between the
vertices at (2, 1) and (2, -2) for B is 3.
4
or 2.
The scale factor is −
2
3
1
or −
.
The scale factor is −
6
2
Exercises
Determine whether the dilation from A to B is an enlargement or a reduction.
Then find the scale factor of the dilation.
1.
y
"
2.
y
"
#
#
1
reduction; −
enlargement; 2
3.
4.
y
"
y
"
x
0
x
0
x
0
2
#
x
0
#
4
reduction; −
1
reduction; −
5
Chapter 7
2
37
Glencoe Geometry
Lesson 7-6
Similarity Transformations
NAME
7-6
DATE
PERIOD
Study Guide and Intervention (continued)
Similarity Transformations
Verify Similarity You can verify that a dilation produces a similar figure by comparing
all corresponding sides and angles. For triangles, you can also use SAS Similarity.
Example
Graph the original figure and its dilated image. Then verify that
the dilation is a similarity transformation.
a. original: A(-3, 4), B(2, 4), C(-3, -4) b. original: G(-4, 1), H(0, 4), J(4, 1)
image: D(1, 0), E(3.5, 0), F(1, -4)
image: L(-2, 1.5), M(0, 3), N(2, 1.5)
Graph each figure. Since ∠A and ∠D
are both right angles, ∠A ∠D. Show
that the lengths of the sides that
include ∠A and ∠D are proportional to
prove similarity by SAS.
Use the distance formula to find the length of
each side.
"
y
) y
(
-
.
#
0
'
AC
8
5
AB
−
=−
= 2 and −
=−
= 2,
DE
x
2.5
Since the lengths of the sides that
include ∠A and ∠D are proportional,
ABC ∼ DEF by SAS similarity.
or 5
42 + 32 = √25
GH = √
2
2
HJ = √4
+ 3 = √
25 or 5
2
2
GJ = √
8 + 0 = √
64 or 8
2
2
LM = √
2 + 1.5 = √
6.25 or 2.5
2
2
MN = √
2 + 1.5 = √
6.25 or 2.5
2
2
or 4
LN = √
4 + 0 = √16
Find and compare the ratios of corresponding
sides.
GH
5
HJ
5
−
=−
or 2, −
=−
or 2,
2.5
2.5
LM
MN
GJ
8
−
=−
or 2
4
LN
GH
HJ
GJ
=−
=−
, GHJ ∼ LMN by
Since −
LM
MN
LN
SSS similarity.
Exercises
Graph the original figure and its dilated image. Then verify that the dilation is a
similarity transformation.
See students’ work.
1. A(-4, -3), B(2, 5), C(2, -3);
D(-2, -2), E(1, 3), F(1, -2)
Chapter 7
2. P(-4, 1), Q(-2, 4), R(0, 1);
W(1, -1.5), X(2, 0), Y(3, -1.5)
38
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Use the coordinate grid to find the
lengths of vertical segments AC and
DF and horizontal segments AB and
DE.
4
DF
AC
AB
.
so − = −
DF
DE
+
& x
0 %
$
/