GUIDED PRACTICE
✓
Concept Check ✓
CP
Vocabulary Check
? to its preimage. similar
1. In a dilation every image is scale factor of the dilation shown to
CP ‘
1
be ᎏᎏ. What did Katie do wrong?
2
than ᎏᎏ.
✓
3. Enlargement; the scale
factor is greater than 1.
(Also, it is apparent that
the image is larger than
the preimage.)
BASIC
P⬘
4
P
2
3. Is the dilation shown a reduction or
Skill Check
ASSIGNMENT GUIDE
2. ERROR ANALYSIS Katie found the
2. She found ᎏᎏ rather
CP ‘
CP
3 APPLY
Day 1: pp. 509–512 Exs. 8–22
even, 35, 36
Day 2: pp. 509–513 Exs. 9–23 odd,
32, 34, 38–44, Quiz 3
Exs. 1–8
C
an enlargement? How do you know?
See margin.
¤PQR is mapped onto ¤P§Q§R§ by a dilation with center C. Complete the
statement.
AVERAGE
4. ¤PQR is (similar, congruent) to ¤P§Q§R§. similar
CP§
4
5. If ᎏᎏ = ᎏᎏ, then ¤P§Q§R§ is (larger, smaller) than ¤PQR, and the dilation is
CP
3
(a reduction, an enlargement). larger; enlargement
Day 1: pp. 509–512 Exs. 8–22
even, 35, 36
Day 2: pp. 509–513 Exs. 9–23 odd,
27–32, 34, 38–44, Quiz 3
Exs. 1–8
ADVANCED
Use the following information to draw a
dilation of rectangle ABCD.
Day 1: pp. 509–512 Exs. 8–22
even, 35, 36
Day 2: pp. 509–513 Exs. 9–23 odd,
27–32, 34, 37–44, Quiz 3
Exs. 1–8
y
6. Draw a dilation of rectangle ABCD on a
coordinate plane, with A(3, 1), B(3, 2.5),
C(5, 2.5), and D(5, 1). Use the origin as the
center and use a scale factor of 2. See margin.
1
B
C
A
D
BLOCK SCHEDULE
1
x
7. Is ABCD ~ A§B§C§D§? Explain your answer.
Yes; Sample answer: a preimage and its image after a dilation are ~.
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 818.
IDENTIFYING DILATIONS Identify the dilation and find its scale factor.
8.
9.
6
P
14
P’
C
C
P
9
24
P’
Reduction; the dilation has center C
Enlargement; the dilation has
3
8
and scale factor }}.
center C and scale factor }}.
7
3
FINDING SCALE FACTORS Identify the dilation, and find its scale factor.
Then, find the values of the variables.
10.
11. A
J’
STUDENT HELP
16
HOMEWORK HELP
Example 1: Exs. 8–11,
20–23
Example 2: Exs. 12–15
Example 3: Exs. 24–26,
33
K’ y
32
28
M’
M
x
K
14
B
25
J
10
C
x
A’
8
D’
C
B’
8
E’
y
10
Enlargement; the dilation has
D
E
z
center C and scale factor
Reduction; the dilation has center C and
2
}}; x = 8, y = 16.
2
1
scale factor }}; x = y = 20, z = 25.
5
8.7 Dilations
pp. 509–512 Exs. 8–22 even,
35, 36 (with 8.6)
pp. 509–513 Exs. 9–23 odd,
27–32, 34, 38–44, Quiz 3 Exs. 1–8
(with Ch. Rev.)
EXERCISE LEVELS
Level A: Easier
8, 9
Level B: More Difficult
10–36
Level C: Most Difficult
37
HOMEWORK CHECK
To quickly check student understanding of key concepts, go
over the following exercises:
Exs. 8, 10, 14, 18, 20, 22, 32. See
also the Daily Homework Quiz:
• Blackline Master (Chapter 9
Resource Book, p. 11)
•
Transparency (p. 63)
6. See Additional Answers beginning
on page AA1.
509
509
TEACHING TIPS
EXERCISE 16 Students should
use the Pythagorean Theorem and
the fact that EG = 2PG and GF = 2GQ
when finding the ratio of EF to PQ.
The result should not be surprising,
since the preimage and image of a
dilation are similar figures.
冉
L’ 冉
14.D’ 冉
F’ 冉
12.J’
冊
冊
冊
冊
DILATIONS IN A COORDINATE PLANE Use the origin as the center of the
冉 冊
冉
冊
冉 冊
冉
冊
dilation and the given scale factor to find the coordinates of the vertices of
1 1
1
º2}}, 1}} , K’ 1, 1}} ,
the image of the polygon.
2 2
2
1
1
1
1, º1}} , M’ º2}}, º1}} 12. k = ᎏ1ᎏ
13. k = 2 P’(6, 10), Q’(8, 0), R’(2, 2)
2
2
2
2
2 1
2
y
y
º1}}, 1}} , E’ 1, }} ,
3 3
3
P
K
J
2
1
1}}, º1 , G’ º1, º1}}
3
3
1
x
1
1
R
œ
x
1
L
M
15. k = 4 S’(º20, 8), T’(º12, 16), U’(º4, 4),
V’(º12, º4)
14. k = ᎏ31ᎏ See margin.
y
D
y
E
T
1
2
S
x
1
U
1
F
x
V
G
16. COMPARING RATIOS Use the triangle shown. Let
Æ
Æ
P and Q be the midpoints of the sides EG and FG,
respectively. Find the scale factor and the center of
the dilation that enlarges ¤PQG to ¤EFG. Find
the ratio of EF to PQ. How does this ratio compare
to the scale factor? 2, G; 2:1; same as the scale factor
E
G
F
CONSTRUCTION Copy ¤DEF and points G and H as shown. Then, use
a straightedge and a compass to construct the dilation. 17–19. Check drawings.
D
17. k = 3; Center: G
1
18. k = ᎏᎏ; Center: H
2
19. k = 2; Center: E
H
G
F
E
SIMILAR TRIANGLES The red triangle is the image of the blue triangle
after a dilation. Find the values of the variables. Then find the ratio of
their perimeters.
20.
21.
C
r
12
t
30
9
510
510
Chapter 8 Similarity
x
C
9
15
r = 20, t = 18; 5:3
9.6
12
x = 7.2, y = 6.3; 3:4
y
8.4
INT
STUDENT HELP
NE
ER T
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for help with problem
solving in Exs. 22 and 23.
IDENTIFYING DILATIONS ¤ABC is mapped onto ¤A§B§C§ by a dilation. Use
the given information to sketch the dilation, identify it as a reduction or an
enlargement, and find the scale factor. Then find the missing lengths.
22. In ¤ABC, AB = 6, BC = 9, and AC = 12. In ¤A§B§C§, A§B§ = 2. Find the
Æ
Æ
lengths of B§C§ and A§C§. Check drawings; reduction; k = }1}; 3, 4
3
23. In ¤ABC, AB = 5 and BC = 7. In ¤A§B§C§, A§B§ = 20 and A§C§ = 36. Find
Æ
Æ
the lengths of AC and B§C§. Check drawings; enlargement; k = 4; 9, 28
FLASHLIGHT IMAGE In Exercises 24–26, use the following information.
STUDENT HELP NOTES
Homework Help Students can
find help for Exs. 22 and 23 at
www.mcdougallittell.com. The
information can be printed out for
students who don’t have access
to the Internet.
24. See Additional Answers beginning on page AA1.
You are projecting images onto a wall with a flashlight. The lamp of the
flashlight is 8.3 centimeters away from the wall. The preimage is imprinted
onto a clear cap that fits over the end of the flashlight. This cap has a diameter
of 3 centimeters. The preimage has a height of 2 centimeters, and the lamp of
the flashlight is located 2.7 centimeters from the preimage.
24. Sketch a diagram of the dilation.
See margin.
25. Find the diameter of the circle of
light projected onto the wall from
the flashlight. about 9.2 cm
26. Find the height of the image
projected onto the wall. about 6.1 cm
ENLARGEMENTS In Exercises 27 and 28, use the following information.
By adjusting the distance between the negative and the enlarged print in the
photographic enlarger shown, you can make prints of different sizes.
In the diagram shown, you want the enlarged print to be 7 inches wide (A§B§).
The negative is 1 inch wide (AB), and the distance between the light source and
the negative is 1.25 inches (CD).
C
light
source
A
B
D
negative
print
A’
D’
B’
27. What is the scale factor of the enlargement? 7:1
28. What is the distance between the light source and the enlarged print? 8.75 in.
DIMENSIONS OF PHOTOS Use the diagram from Exercise 27 to
determine the missing information.
CD
CD §
AB
A §B§
29.
1.2 in.
7.2 in.
0.8 in.
?
30.
?
14 cm
2 cm
12 cm
about 2.33 cm
31.
2 in.
10 in.
?
8.5 in.
1.7 in.
4.8 in.
8.7 Dilations
511
511
FOCUS ON
CAREERS
CAREER NOTE
EXERCISE 34 Additional
information about architectural
rendering is available at
www.mcdougallittell.com.
32.
LOGICAL REASONING Draw any triangle, and label it ¤PQR. Using a
scale factor of 2, draw the image of ¤PQR after a dilation with a center
outside the triangle, with a center inside the triangle, and with a center on
the triangle. Explain the relationship between the three images created.
33.
Writing Use the information about shadow puppet theaters from
See margin.
Example 3, page 508. Explain how you could use a shadow puppet theater to
help another student understand the terms image, preimage, center of
dilation, and dilation. Draw a diagram and label the terms on the diagram.
33. Sample answer:
S
34.
C
L
P
RE
ARCHITECTURAL
RENDERING An
architectural renderer uses
techniques like the one
shown in Exercise 34 to
create three dimensional
drawings of buildings and
other structures.
INT
I would show the student this
figure and explain that L represents the light source, CP the
height of the puppet, and SH the
height of the image. I would tell
the student that a dilation transforms a figure in such a way that
the original figure (or preimage)
and the figure after the dilation
(the image) are similar. In the
figure, the center of the dilation
is L and the scale factor is the
ratio of LS to LC.
FE
L
AL I
H
See margin.
PERSPECTIVE DRAWING Create a perspective drawing by following the
given steps. Check drawings.
NE
ER T
CAREER LINK
www.mcdougallittell.com
Test
Preparation
32.The ◊ are £. All 3 ◊ are
~ to ¤PQR, so all the
corresp. Å are £. The
scale factor of each
dilation is 2, so each side
of every image ¤ is twice
as long as the corresp.
side of ¤PQR. Then the
corresp. sides of the 3 ◊
are £ and the 3 ◊ are £
by the def. of £ ◊.
1
Draw a horizontal line
across the paper, and
choose a point on this
line to be the center of
the dilation, also called
the vanishing point.
Next, draw a polygon.
2
Draw rays from the
vanishing point to all
vertices of the polygon.
Draw a reduction of
the polygon by locating
image points on the
rays.
A
¡
B
¡
enlargement; k = 2
D
1
enlargement; k = ᎏᎏ
3
C
¡
reduction; k = ᎏᎏ
D
¡
E
¡
1
reduction; k = ᎏᎏ
2
1
3
4
D’
2
C
E’
F’
• Practice Levels A, B, and C
(Chapter 8 Resource Book, p. 101)
• Reteaching with Practice
(Chapter 8 Resource Book, p. 104)
•
See Lesson 8.7 of the
Personal Student Tutor
For more Mixed Review:
•
Search the Test and Practice
Generator for key words or
specific lessons.
512
★ Challenge
3%
1
33ᎏᎏ%
3
B
¡
E
¡
12%
E
F
reduction; k = 3
36. MULTIPLE CHOICE In the diagram shown,
A
¡
D
¡
For Lesson 8.7:
Connect the
preimage to the
image by darkening
the segments
between them.
Erase all hidden
lines.
35. MULTIPLE CHOICE Identify the dilation shown as an enlargement or
reduction and find its scale factor. C
the center of the dilation of ⁄JKLM is point
C. The length of a side of ⁄J§K§L§M§ is what
percent of the length of the corresponding
side of ⁄JKLM? D
ADDITIONAL PRACTICE
AND RETEACHING
3
C
¡
C
K’
J’
M’
K
2
L’
10
20%
300%
L
M
6
37. CREATING NEW IMAGES A polygon is reduced by a dilation with center C
1
and scale factor ᎏᎏ. The image is then enlarged by a dilation with center C and
k
scale factor k. Describe the size and shape of this new image.
The new image is the original figure.
512
J
Chapter 8 Similarity
MIXED REVIEW
4 ASSESS
USING THE PYTHAGOREAN THEOREM Refer to the
triangle shown to find the length of the missing
side by using the Pythagorean Theorem.
(Review 1.3 for 9.1)
DAILY HOMEWORK QUIZ
c
a
Transparency Available
1. Identify the dilation and find
its scale factor. Then find the
values of the variables.
b
38. a = 5, b = 12 c = 13 39. a = 8, c = 2兹6
苶5苶 b = 14
40. b = 2, c = 5兹5
苶 a = 11 41. b = 1, c = 兹5苶0苶 a = 7
R
42. Find the geometric mean of 11 and 44. (Review 8.2 for 9.1) 22
x
DETERMINING SIMILARITY Determine whether the triangles can be proved
similar or not. Explain your reasoning. (Review 8.4 and 8.5)
C
T´
B
J
44.
K
L
CA
LJ
QUIZ 3
3
5
T
Yes; ™P £ ™T and ™R £ ™S by the
Alternate Interior Angles Thm. (Vertical Å
PQR and TQS are also £.) Then ¤PQR ~
¤TQS by the AA Similarity Post.
CB
LK
™C £ ™L and }} = }}, so the ◊
are ~ by the SAS Similarity Thm.
S
scale factor }}; x = 10.8, y = 9
R
Yes; Sample answer:
S´
The dilation has center C and
S
q
C
20
12 T
P
34ⴗ
34ⴗ
18
15
y
43. A
R´
Self-Test for Lessons 8.6 and 8.7
Use the figure to complete the proportion.
A
(Lesson 8.6)
AC
AB
1. ᎏᎏ = ᎏᎏ BD
CE
?
?
BD
2. ᎏᎏ = ᎏᎏ CE
CG
BF
EG
DF
3. ᎏᎏ = ᎏᎏ AF
AG
?
?
GA
4. ᎏᎏ = ᎏᎏ FA
DA
EA
B
D
C
Use the origin as the center of the
dilation, the given scale factor,
and the given coordinates of the
vertices of the preimage polygon
to find the coordinates of the vertices of the image polygon.
2. 䉭XYZ, with scale factor 5 and
vertices X(3, 1), Y(6, 4), and
Z(8, –1) Xⴕ(15, 5), Yⴕ(30, 20),
Z ⴕ(40, –5)
2
3. ⵥPQRS with scale factor ᎏᎏ
3
and vertices P(–6, 3), Q(3, 3),
R(3, –6), and S(–6, –6)
E
F
G
Pⴕ(–4, 2), Qⴕ(2, 2), Rⴕ(2, –4),
Sⴕ(–4, –4)
In Exercises 5 and 6, identify the dilation and find its scale factor. (Lesson 8.7)
5.
6.
P’
5
EXTRA CHALLENGE NOTE
C
P
Challenge problems for
Lesson 8.7 are available in
blackline format in the Chapter 8
Resource Book, p. 108 and at
www.mcdougallittell.com.
21
5
H’
C
The dilation is an enlargement with
center C and scale factor 2.
42
H
The dilation is a reduction with
1
center C and scale factor }}.
CJ§ 3 5
7. ¤JKL is mapped onto ¤J§K§L§ by a dilation, with center C. If ᎏᎏ = ᎏᎏ,
CJ
6
then the dilation is (a reduction, an enlargement) and ¤JKL is (larger,
smaller) than ¤J§K§L§. (Lesson 8.7) reduction, larger
8.
ENLARGING PHOTOS An 8 inch by 10 inch photo is enlarged to produce
1
an 18 inch by 22 ᎏᎏ inch photo. What is the scale factor? (Lesson 8.7) }9}
2
4
8.7 Dilations
Additional Test Preparation Sample answer:
1. Check students’ drawings. After drawing the triangle,
draw a line through C and each vertex of the triangle.
Use a compass to mark a point three times as far from C
ADDITIONAL TEST
PREPARATION
1. WRITING Draw a triangle and
a point C, outside the triangle.
Describe how to draw the image
of the triangle after a dilation
with scale factor 3. See left.
513
as each original vertex and on the line between C and
the vertex. Join these three points to form the image.
513