3 APPLY
PRACTICE AND APPLICATIONS
WRITING SIMILARITY STATEMENTS Use the information given to list all
pairs of congruent angles and write the statement of proportionality for
the figures.
STUDENT HELP
ASSIGNMENT GUIDE
BASIC
Day 1: pp. 476–477 Exs. 8–38
Day 2: pp. 477–479 Exs. 39–42,
45–49, 53–67, Quiz 1
Exs. 1–10
AVERAGE
Day 1: pp. 476–477 Exs. 8–38
Day 2: pp. 477–479 Exs. 39–42,
45–49, 53–67, Quiz 1
Exs. 1–10
ADVANCED
Day 1: pp. 476–477 Exs. 8–38
Day 2: pp. 477–479 Exs. 39–42,
45–49, 50–67, Quiz 1
Exs. 1–10
BLOCK SCHEDULE
pp. 476–477 Exs. 8–38 (with 8.2)
pp. 477–479 Exs. 39–42, 45–49,
53–67, Quiz 1 Exs. 1–10 (with 8.4)
Extra Practice
to help you master
skills is on p. 817.
DE
10. ™Q £ ™A, ™R £ ™B,
™S £ ™C, ™T £ ™D,
QR
AB
ST
TU
QU
}} = }} = }}
CD
DE
AE
RS
BC
™U £ ™E; }} = }} =
11. Yes; both figures are
rectangles, so all 4 Å are
AB
FG
AD
7
}} = }}.
FE
4
BC
GH
13. No; m™B = 90° and
m™Q = 88°, so corresp.
Å are not £.
14. No; m™K = 90° and
m™Q = 88°, so corresp.
EXERCISE LEVELS
Level A: Easier
8–10
Level B: More Difficult
11–49
Level C: Most Difficult
50–52
HOMEWORK CHECK
To quickly check student understanding of key concepts, go over
the following exercises: Exs. 10,
12, 18, 20, 28, 30, 38, 42, 46. See
also the Daily Homework Quiz:
• Blackline Master (Chapter 8
Resource Book, p. 49)
•
Transparency (p. 58)
DETERMINING SIMILARITY Decide whether the quadrilaterals are similar.
Explain your reasoning. 11–14. See margin.
A
B
CD
HE
7
3.5
}} ≠ }} .
5
3
6
3
Å are not £ and }} ≠ }},
5
3
E
7
D
L
4
C
H 2
q
88ⴗ
P
3
5
6
R
G
3
J
12. ABCD and JKLM
13. ABCD and PQRS
14. JKLM and PQRS
DETERMINING SIMILARITY Decide whether the polygons are similar. If so,
write a similarity statement.
15.
E
A
F
B
16. T
no
U
W
D
4
C
5
H
6
A
K 118ⴗ
R
q
C
J
STUDENT HELP
yes; ¤XYZ ~ ¤CAB
USING SIMILAR POLYGONS PQRS ~ JKLM.
HOMEWORK HELP
19. Find the scale factor of PQRS to JKLM. 4:5
105ⴗ
10 P
K
q
15
12
L
25
R
w
10
x
21. Find the values of w, x, and y. 20, 12.5, 20
22. Find the perimeter of each polygon.
JKLM: 72.5; PQRS: 58
23. Find the ratio of the perimeter of PQRS to
S
M
20. Find the scale factor of JKLM to PQRS. 5:4
Chapter 8 Similarity
G
75ⴗ
Y
5
3
14
65ⴗ
30
20
the perimeter of JKLM. }4}
H
L
8
9
4
V
6
18.
B
6.75
Z
F
G
4.5
12
E
4
5
4
no
476
S
M
11. ABCD and FGHE
17. X
Example 1: Exs. 8–10
Example 2: Exs. 11–18
Example 3: Exs. 19–30,
43, 44
Example 4: Exs. 19–30,
46–48
Example 5: Exs. 39–42
476
K
F
3.5
so lengths of corresp.
sides are not proportional.
15. yes; Sample answers:
ABCD ~ EFGH,
ABCD ~ FEHG
DF
JK
KL
JM
LM
}} = }} = }} = }}
WX
XY
WZ
YZ
10. QRSTU ~ ABCDE
£ and }} = }} = }} =
12. No; lengths of corresp.
sides are not proportional;
EF
} = }} = }}
8. ¤DEF ~ ¤PQR ™D £ ™P, ™E £ ™Q, ™F £ ™R; }
PQ
QR
PR
™J £ ™W, ™K £ ™X, ™L £ ™Y, ™M £ ™Z;
9. ⁄JKLM ~ ⁄WXYZ
J
y
M
P
16
S
USING SIMILAR POLYGONS ⁄ABCD ~ ⁄EFGH.
F
E
24. Find the scale factor of ⁄ABCD to ⁄EFGH. 2:1
H
Æ
25. Find the length of EH. 2
3
G
B
A
26. Find the measure of ™G. 150°
4
30ⴗ
27. Find the perimeter of ⁄EFGH. 10
6
D
C
1
2
28. Find the ratio of the perimeter of ⁄EFGH to the perimeter of ⁄ABCD. }}
DETERMINING SIMILARITY Decide whether the polygons are similar. If so,
find the scale factor of Figure A to Figure B.
18
29. no
18
30.
8
110ⴗ
A
14
INT
STUDENT HELP
NE
ER T
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for help with problem
solving in Exs. 31–38.
44. No; the ratio of length
4
3
to width = }} for a
standard screen and
16
}} for a digital screen.
9
4
length
Then }} = }}
16
length
width
3
and }} = }}, and
width
9
4
3
}} ≠ }}.
16
9
4
18
6
LOGICAL REASONING Tell whether the polygons are always,
sometimes, or never similar.
31. Two isosceles triangles sometimes
35. Two squares
32. Two regular polygons sometimes
sometimes 34. Two rhombuses sometimes
33. Two isosceles trapezoids
36. An isosceles and a scalene triangle
never
37. Two equilateral triangles always
38. A right and an isosceles triangle
sometimes
xy USING ALGEBRA The two polygons are similar. Find the values of x and y.
always
39.
40.
15
xⴚ2
60ⴗ
10
6
18
8
11, 9
41.
FOCUS ON
APPLICATIONS
Homework Help Students
can find help for Exs. 31–38
at www.mcdougallittell.com.
The information can be printed
out for students who don’t have
access to the Internet.
B
A
70ⴗ
30
STUDENT HELP NOTES
yes; 3:2
B
COMMON ERROR
EXERCISES 16–18 Students
often have difficulty determining
the corresponding sides of similar
figures when one figure is rotated.
To avoid this problem, students can
trace one of the figures and rotate
the tracing to make the two similar
figures easier to compare.
y ⴗ 12
22
yⴙ3
xⴙ4
2
3
10 }}, 120
42.
27
(y ⴚ 73)ⴗ
21
39
61ⴗ
x
116ⴗ
18
xⴚ6
4
116ⴗ
5
6
y
1
3
39 }}, 23 }}
7
7
7.5, 166
TV SCREENS In Exercises 43 and 44, use the following information.
RE
FE
L
AL I
Television screen sizes are based on the length of the diagonal of the screen. The
aspect ratio refers to the length to width ratio of the screen. A standard 27 inch
analog television screen has an aspect ratio of 4:3. A 27 inch digital television
screen has an aspect ratio of 16:9.
DIGITAL
TELEVISION
screens contain over 6 times
as many pixels (the tiny dots
that make up the picture) as
standard analog screens.
43. Make a scale drawing of each television screen. Use proportions and
the Pythagorean Theorem to calculate the lengths and widths of the
screens in inches. Check scale drawings; analog tv: 21.6 in by 16.2 in.;
digital tv: about 23.5 in. by 13.2 in.
44. Are the television screens similar? Explain. See margin.
8.3 Similar Polygons
477
477
45. ABCD Í EFGH with scale factor
AB BC C D AD 1
1:k, so }} = }} = }} = }} = }}.
EF
FG
GH
EH
k
Then EF = k ⴢ AB, FG = k ⴢ BC,
GH = k ⴢ CD, and EH = k ⴢ AD, so
perimeter of ABCD
}} =
perimeter of EFGH
AB + BC + CD + AD
}} =
EF + FG + GH + EH
AB + BC + CD + AD
}}}} =
k ⴢ AB + k ⴢ BC + k ⴢ CD + k ⴢ AD
AB + BC + CD + AD
}}} =
k (AB + BC + CD + AD )
1 AB
}} = }}.
k EF
49.a. Figure 1: AB = 4.2,
BC = 2.1, CD = 3.5,
DE = 1.4, EA = 2.8
b. See margin for graph;
10
7
y = }}x
10
7
c. }}; it is the reciprocal
of the scale factor.
PROOF Prove Theorem 8.1 for two
similar rectangles. See margin.
E
A
GIVEN 䉴 ABCD ~ EFGH
perimeter of ABCD
perimeter of EFGH
AB
EF
D
PROVE 䉴 ᎏᎏ = ᎏᎏ
L
kpL
H
C
G
46. SCALE The ratio of the perimeter of WXYZ to the perimeter
of QRST is 7.5: 2. Find the scale factor of QRST to WXYZ. 2:7.5 or 4:15
¤FGH is 2:5. The perimeter of ¤FGH is 28 inches. Find the perimeter
of ¤CDE. 11.2 in.
48. SCALE The perimeter of ⁄PQRS is 94 centimeters. The perimeter of
⁄JKLM is 18.8 centimeters, and ⁄JKLM ~ ⁄PQRS. The lengths of the
sides of ⁄PQRS are 15 centimeters and 32 centimeters. Find the scale factor
of ⁄PQRS to ⁄JKLM, and the lengths of the sides of ⁄JKLM.
Test
Preparation
y
94:18.8 or 5:1; 3 cm and 6.4 cm
49. MULTI-STEP PROBLEM Use the similar figures shown. The scale factor of
Figure 1 to Figure 2 is 7:10. See margin.
D
AB
BC
CD
DE
EA
Figure 1
?
?
?
?
?
Figure 2
6.0
3.0
5.0
2.0
4.0
E
E
C
A
B
Figure 1
2
D
5
C
4
A
3
6
Figure 2
B
b. Graph the data in the table. Let x represent the length of a side in Figure 1
1
1
and let y represent the length of the corresponding side in Figure 2.
Determine an equation that relates x and y.
x
c. ANALYZING DATA The equation you obtained in part (b) should be
50–52. See Additional Answers
beginning on page AA1.
★ Challenge
linear. What is its slope? How does its slope compare to the scale factor?
TOTAL ECLIPSE Use the following information in Exercises 50–52.
From your perspective on Earth during a total eclipse of the sun, the moon is
directly in line with the sun and blocks the sun’s rays. The ratio of the radius of
the moon to its distance to Earth is about the same as the ratio of the radius of the
sun to its distance to Earth. 50–52. See margin.
Distance between Earth and the moon: 240,000 miles
Distance between Earth and the sun: 93,000,000 miles
Radius of the sun: 432,500 miles
50. Make a sketch of Earth, the moon, and the sun
during a total eclipse of the sun. Include the
given distances in your sketch.
ADDITIONAL PRACTICE
AND RETEACHING
51. Your sketch should contain some similar
For Lesson 8.3:
• Practice Levels A, B, and C
(Chapter 8 Resource Book, p. 38)
• Reteaching with Practice
(Chapter 8 Resource Book, p. 41)
•
See Lesson 8.3 of the
Personal Student Tutor
For more Mixed Review:
Search the Test and Practice
Generator for key words or
specific lessons.
478
kpw
w
a. Copy and complete the table.
•
F
B
47. SCALE The ratio of one side of ¤CDE to the corresponding side of similar
49a. 4.2; 2.1; 3.5; 1.4; 2.8
49b.
45.
triangles. Use the similar triangles in your
sketch to explain a total eclipse of the sun.
52. Write a statement of proportionality for the
EXTRA CHALLENGE
www.mcdougallittell.com
478
similar triangles. Then use the given distances
to estimate the radius of the moon.
Chapter 8 Similarity
MIXED REVIEW
4 ASSESS
FINDING SLOPE Find the slope of the line that passes through the given
points. (Review 3.6 for 8.4)
2
º}}
53. A(º1, 4), B(3, 8) 1
54. P(0, º7), Q(º6, º3) 3 55. J(9, 4), K(2, 5) º}1}
DAILY HOMEWORK QUIZ
7
56. L(º2, º3), M(1, 10) }13} 57. S(–4, 5), T(2, º2) º}7}
6
3
58. Y(º1, 6), Z(5, º5) º}11}
6
FINDING ANGLE MEASURES Find the value of x. (Review 4.1 for 8.4)
59. A
60. B
49
83ⴗ
xⴗ
5x ⴗ
61. 9
A
B
C
3x ⴗ
(9x ⴚ 1)ⴗ
C
A
(6x ⴚ 6)ⴗ
8
7
41ⴗ
Transparency Available
Are the polygons similar? If so,
write a similarity statement.
S
R
M
1.
8
8
105ⴗ
L
B
C
Q
N
no
2.
12
D
C
SOLVING PROPORTIONS Solve the proportion. (Review 8.1)
x
6
62. ᎏᎏ = ᎏᎏ 2
9
27
4
b
65. ᎏᎏ = ᎏᎏ
13
8
2
4
63. ᎏᎏ = ᎏᎏ 38
19
y
32
}}
13
11
9
66. ᎏᎏ = ᎏᎏ
x+2
x
H
5
25
64. ᎏᎏ = ᎏᎏ 120
24
z
3x + 7
4x
67. ᎏᎏ = ᎏᎏ
5
6
9
QUIZ 1
8
16
9
16
75° 12
12
75°
21
G
12
A
105°
B
E
105°
9
F
yes; Sample answer:
ABCD Í FEHG
Self-Test for Lessons 8.1–8.3
In Exs. 3–5, 䉭KLM ⬃ 䉭PQR.
Q
Solve the proportions. (Lesson 8.1)
p
2
1. ᎏᎏ = ᎏᎏ 10
15
3
4
16
3. ᎏᎏ = ᎏᎏ }24}
2x º 6
x 7
5
20
2. ᎏᎏ = ᎏ 28
7
d
K
8
Find the geometric mean of the two numbers. (Lesson 8.2)
4. 7 and 63
5. 5 and 11
21
兹55
苶 ≈ 7.42
L
6. 10 and 7
兹70
苶 ≈ 8.37
In Exercises 7 and 8, the two polygons are similar. Find the value of x. Then
find the scale factor and the ratio of the perimeters. (Lesson 8.3)
7. 3; 1:2; }1}
3
2
30; 3:2; }}
8.
2
6
8
x
20
冉
5
}} ≈ 2.22,
2.25
7
}} ≈ 2.15
3.25
冊
R
3. Find the scale factor. 2:3
4. Find the length of P苶Q
苶. 12
5. Find the measure of ⬔Q. 30°
EXTRA CHALLENGE NOTE
Challenge problems for
Lesson 8.3 are available in
blackline format in the Chapter 8
Resource Book, p. 45 and at
www.mcdougallittell.com.
You are ordering your school pictures. You decide to order one 8 ª 10 (8 inches
冉 14
by 10 inches), two 5 ª 7’s (5 inches by 7 inches), and 24 wallets 2ᎏᎏ inches by
1
4
M
x
COMPARING PHOTO SIZES Use the following information. (Lesson 8.3)
9. None are exactly
similar, but the 5 ª 7
and wallet sizes are
very nearly similar.
45°
10
45
3
xⴙ1
15
P
105°
冊
3ᎏᎏ inches .
ADDITIONAL TEST
PREPARATION
1. OPEN ENDED Draw two similar figures. Explain why they are
similar and write the similarity
statement. Check students’ work.
9. Are any of these sizes similar to each other?
10. Suppose you want the wallet photos to be similar to the 8 ª 10 photo. If the
1
1
wallet photo were 2ᎏᎏ inches wide, how tall would it be? 3}8} in.
2
8.3 Similar Polygons
479
ADDITIONAL RESOURCES
An alternative Quiz for Lessons
8.1–8.3 is available in the Chapter 8
Resource Book, p. 46.
479