Page 1 of 6
7.6
Goal
Identify and draw
dilations.
Key Words
• dilation
Dilations
Geo-Activity
Drawing a Dilation
1 Draw a triangle. Label it TPQR.
●
Choose a point C outside the
triangle.
2 Use a straightedge to draw
●
lines from C through P, Q,
and R.
• reduction
• enlargement
P
P
Œ
Œ
R
C
R
C
3 Measure CP
&* and locate a point
●
&*( such that CP9 5 2 p CP.
P9 on CP
Locate Q9 and R9 the same way.
4 Connect the points P9, Q9,
●
and R9 to form TP9Q9R9.
P9
P9
Œ9
Œ9
P
P
Œ
C
R
Œ
R9
R
C
R9
P9R9 R9Q9
Q9P9
}, }}, and }}. Then show that TPQR is similar
5 Calculate the ratios }
●
PR RQ
QP
to TP9Q9R9.
6 What is the scale factor of TP9Q9R9 to TPQR ? How does it compare to
●
CP 9
CP
the ratio }} ?
Student Help
LOOK BACK
To review other types
of transformations, see
pp. 152 and 282.
A dilation is a transformation with
center C and scale factor k that maps
each point P to an image point P9 so
&*( and CP9 5 k p CP.
that P9 lies on CP
As you saw in the Geo-Activity above,
a dilation maps a figure onto a similar
figure, called the image.
k p CP
P9
image
P
point
C
center of
dilation
7.6
Dilations
393
Page 2 of 6
Types of Dilations If the image is smaller than the original figure,
then the dilation is a reduction . If the image is larger than the
original figure, then the dilation is an enlargement .
Visualize It!
EXAMPLE
original
Tell whether the dilation is a reduction or an enlargement.
a.
1
Identify Dilations
b. X
P9
X9
P
image
R9
C
Œ
Œ9
For transformations in
this book, the original
figure is blue and the
image is red.
Y
Y9
Z9
C
R
Z
Solution
a. The dilation is an enlargement because the image (T P9Q9R9)
is larger than the original figure (T PQR).
b. The dilation is a reduction because the image (T X9Y9Z9) is
smaller than the original figure (T XYZ ).
Student Help
STUDY TIP
Scale Factor The scale factor of a dilation is the ratio of CP9 to CP.
This is also equal to the scale factor of T P9Q9R9 to T PQR.
P
If you know the scale
factor k of a dilation,
you can tell the type of
dilation.
P9
Œ
Œ9
C
If k < 1, then the dilation
is a reduction.
R9
R
If k > 1, then the dilation
is an enlargement.
CP 9
CP
P 9Q 9
PQ
scale factor 5 k 5 }} 5 }}
EXAMPLE
2
Find Scale Factors
Find the scale factor of the dilation.
a.
b.
P
3
P9
2
P9
6
P
3
C
C
Solution
Find the ratio of CP9 to CP.
C P9
CP
2
3
a. scale factor 5 }} 5 }}
394
Chapter 7
Similarity
C P9
CP
6
3
b. scale factor 5 }} 5 }} 5 2
Page 3 of 6
Identify Dilations and Find Scale Factors
Tell whether the dilation is a reduction or an enlargement. Then find
the scale factor of the dilation.
1.
2.
C
3.
5
3
P
C
10
C
8 P9
P9
P
P
24
4
P9
EXAMPLE
3
Dilations and Similar Figures
T P9Q9R9 is the image of T PQR after a reduction.
Find the value of x.
Œ
Œ9
x
R
6
Student Help
R9
C
STUDY TIP
Recall that the scale
factor of the dilation
equals the ratio of the
corresponding sides of
the figures.
8
14
P
P9
Solution
C P9
Q9R9
}} 5 }}
CP
QR
Write a proportion.
8
6
}} 5 } }
14
x
Substitute 8 for CP9, 14 for CP, 6 for Q9R9,
and x for QR.
8 p x 5 14 p 6
Cross product property
8x 5 84
Multiply.
x 5 10.5
Divide each side by 8.
Dilations and Similar Figures
The red figure is the image of the blue figure after a dilation.
Find the value of the variable.
4.
5.
m
C
2
8
4
n
3
C
6
20
7.6
Dilations
395
Page 4 of 6
7.6 Exercises
Guided Practice
Vocabulary Check
Skill Check
1. Complete: In a dilation, every image is __?__ to the original figure.
2. Error Analysis Katie found the
scale factor of the dilation shown
1
2
to be }}. What did Katie do wrong?
2
P9
4
P
C
3. Is the dilation shown a reduction or
an enlargement? How do you know?
2
4
1
2
scale factor 5 }} 5 }}
Tell whether the dilation is a reduction or an enlargement.
4.
P9
C
S9
P
5.
Œ9
R9
X9
X
Y9
C
Y
S
Œ
Z
Z9
R
Practice and Applications
Extra Practice
Identifying Dilations Tell whether the dilation is a reduction or
an enlargement. Then find its scale factor.
See p. 688.
6.
7.
P
14
P9
6
C
C
P
9
24
P9
8.
15
396
Chapter 7
Similarity
P
C
9
Homework Help
Example 1: Exs. 6–9,
11–14
Example 2: Exs. 6–9, 15
Example 3: Exs. 11–14,
16
P9
9.
C
28
8
P9
P
10.
Visualize It! Sketch quadrilateral EFGH. Then sketch
a dilation of EFGH with center C and scale factor 3.
Page 5 of 6
Dilations and Similar Figures The red figure is the image of the
blue figure after a dilation. Tell whether the dilation is a reduction
or an enlargement. Then find the value of the variable.
11.
12.
C
9
x
28
15
14
C
n
35
12
13.
14.
y
2
2
3
4
C
m
7
C
28
IStudent Help
ICLASSZONE.COM
HOMEWORK HELP
Extra help with problem
solving in Exs. 15–16 is
at classzone.com
Flashlight Image In Exercises 15 and 16, use the following
information and the diagram below.
The flashlight in the photograph below has a removable clear cap with
a picture of a bug on it. When the flashlight is on, the bug is projected
onto a wall. This situation models a dilation.
15. What is the scale factor of the dilation?
16. Write and solve a proportion to find the
height of the bug projected onto the wall.
2 cm
h
2.5 cm
10 cm
Standardized Test
Practice
17. Multiple Choice The length of a side of J9K9L9M9 is what percent
of the length of the corresponding side of JKLM?
A
X
B
X
3%
C
X
33}}%
X
300%
12%
C
J9
K9
L9
K
J
M9 2
10
1
3
L
D
M
6
7.6
Dilations
397
Page 6 of 6
Mixed Review
Finding Angle Measures Find the measure of a1. (Lesson 4.2)
18.
19.
A
20.
F
J
848
588
358
1148
B
H
C
1
K
708
1
498
1
L
G
Solving Proportions Solve the proportion. (Lesson 7.1)
d
7
12
28
35
10
21. }} 5 }}
Algebra Skills
9
2
7
x11
22. }} 5 }}
t13
12
23. }} 5 }}
Evaluating Expressions Evaluate the expression when x 5 23.
(Skills Review, p. 670)
24. x 2 1 8
15
x
27. 11 2 }}
25. 24x 2 9
26. (x 1 2)(x 2 5)
28. 2x 3
29. 4x 2 1 3x 2 1
Quiz 2
Determine whether the triangles are similar. If they are similar,
write a similarity statement. (Lesson 7.4)
1.
2.
A
8
788
8
14
C
L
J
4
10 K
24
B
F
6 788
P
10.5
R
H
12
P
30
G
Find the value of the variable. (Lesson 7.5)
3. A
x
D
12
F
4.
5. P
E
J
15
C
B
H
K
y
S
G
36
6.
7.
P9
C
10
P
21
5
C
H9
42
H
Chapter 7
Similarity
22
R
Tell whether the dilation is a reduction or an enlargement.
Then find its scale factor. (Lesson 7.6)
398
P
z
10
T