GUIDED PRACTICE
Vocabulary Check
Concept Check
Skill Check
✓
✓
✓
3. False; the ratio of the
lengths of corresp. sides is
1:2, so the ratio of the
areas is 1:4. Doubling the
side length of a square
quadruples the area.
3 APPLY
1. If two polygons are similar with the lengths of corresponding sides in the
? and the ratio of their
ratio of a:b, then the ratio of their perimeters is ?. a:b; a 2 :b 2
areas is ASSIGNMENT GUIDE
BASIC
Day 1: pp. 679–680 Exs. 8–20
even, 23–27
Day 2: pp. 679–682 Exs. 7–21 odd,
28, 29, 34–41, Quiz 1
Exs. 1–7
Tell whether the statement is true or false. Explain.
2. Any two regular polygons with the same number of sides are similar.
True; all corresp. √ are £ and all side lengths are proportional.
3. Doubling the side length of a square doubles the area.
See margin.
In Exercises 4 and 5, the red and blue figures are similar. Find the ratio
(red to blue) of their perimeters and of their areas.
4.
1:3, 1:9
5.
Day 1: pp. 679–680 Exs. 8–20
even, 23–27
Day 2: pp. 679–682 Exs. 7–21 odd,
28, 29, 34–41, Quiz 1
Exs. 1–7
3:2, 9:4
5
1
33
9
ADVANCED
3
6.
AVERAGE
6
4
Day 1: pp. 679–680 Exs. 8–20
even, 23–27
Day 2: pp. 679–682 Exs. 7–21 odd,
22, 28–41, Quiz 1 Exs. 1–7
PHOTOGRAPHY Use the information from Example 2 on page 678 to
find a reasonable cost for a sheet of 4 inch by 5 inch photographic paper.
about $.11
BLOCK SCHEDULE
pp. 679–682 Exs. 7–21, 23–29,
34–41, Quiz 1 Exs. 1–7
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 823.
FINDING RATIOS In Exercises 7–10, the polygons are similar. Find the ratio
(red to blue) of their perimeters and of their areas.
7.
2:1, 4:1
16
5:7, 25:49
8
9.
5
5:6, 25:36
2.5
8.
7
10. 5:3, 25:9
3
12.5
5
7.5
HOMEWORK CHECK
To quickly check student understanding of key concepts, go
over the following exercises:
Exs. 8, 14, 16, 18, 20, 24, 27. See
also the Daily Homework Quiz:
• Blackline Master (Chapter 11
Resource Book, p. 55)
•
Transparency (p. 82)
3
LOGICAL REASONING In Exercises 11–13, complete the statement
using always, sometimes, or never.
? have the same perimeter. sometimes
11. Two similar hexagons STUDENT HELP
HOMEWORK HELP
Example 1: Exs. 7–10,
14–18
Example 2: Exs. 23, 24
Example 3: Exs. 25–28
EXERCISE LEVELS
Level A: Easier
7–13
Level B: More Difficult
14–29
Level C: Most Difficult
30–33
? similar. sometimes
12. Two rectangles with the same area are ? similar. always
13. Two regular pentagons are 14. HEXAGONS The ratio of the lengths of corresponding sides of two similar
hexagons is 2:5. What is the ratio of their areas? 4:25
15. OCTAGONS A regular octagon has an area of 49 m2. Find the scale factor of
this octagon to a similar octagon that has an area of 100 m2. 7:10
11.3 Perimeters and Areas of Similar Figures
679
679
Æ
STUDENT HELP NOTES
Homework Help Students can
find help for Exs. 19–21 at
www.mcdougallittell.com.
The information can be printed
out for students who don’t have
access to the Internet.
Æ Æ
Æ
Æ
18. AB ∞ DC and LK ∞ AB , so
Æ Æ
DC ∞ LK since 2 lines ∞ to
the same line are ∞. Then
™K £ ™C and ™A £ ™J
and all the corresp. √ are
£. The ratio of the lengths
of any 2 corresp. sides is
1
1
}}, so ratio of areas = ᎏᎏ;
3
9
2
137.7 in.
STUDENT HELP
INT
COMMON ERROR
EXERCISE 16 Students may forget to square the numbers in the
ratio of side lengths to find the ratio
of areas. Or, students may not set up
the proportion correctly. Encourage
students to think about their answer
and make sure that it makes sense
in the context of the problem.
17. Æ
Since AB is parallel to
16.
DC , ™A £ ™C and
™B £ ™D by the Alternate
Interior Angles Thm. So
¤ CDE ~¤ ABE by the
17.
AA Similarity Postulate;
98 square units
NE
ER T
8 inches long. Given that the area of ¤ABC is 13.9 square inches, find the
Æ
area of similar triangle ¤DEF whose hypotenuse DF is 20 inches long.
86.875 in.2
FINDING AREA Explain why
18. FINDING AREA Explain why
¤CDE is similar to ¤ABE.
Find the area of ¤CDE.
A
12
3
⁄JBKL ~ ⁄ABCD. The area of
⁄JBKL is 15.3 square inches.
Find the area of ⁄ABCD.
B
A
E
J
12
L
B
4
50⬚
K
5
7
50⬚
D
C
D
15
C
19. SCALE FACTOR Regular pentagon ABCDE has a side length of
6兹5苶 centimeters. Regular pentagon QRSTU has a perimeter of
40 centimeters. Find the ratio of the perimeters of ABCDE to QRSTU. 3兹5苶 :4
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for help with scale
factors in Exs. 19–21.
Æ
RIGHT TRIANGLES ¤ABC is a right triangle whose hypotenuse AC is
20. SCALE FACTOR A square has a perimeter of 36 centimeters. A smaller
square has a side length of 4 centimeters. What is the ratio of the areas of the
larger square to the smaller one? 81:16
22. Let ABCD and EFGH be
similar rectangles with
21. SCALE FACTOR A regular nonagon has an area of 90 square feet. A similar
the lengths of corresp.
nonagon has an area of 25 square feet. What is the ratio of the perimeters of
sides in the ratio a:b. Let
the first nonagon to the second? 3兹10
苶:5
a¬ be the length of ABCD
and aw the width, and let 22.
PROOF Prove Theorem 11.5 for rectangles.
b¬ be the length of EFGH
and bw the width. Then
RUG COSTS Suppose you want to be sure that a large rug is priced
area of ABCD
a¬(aw)
}} = }} = fairly. The price of a small rug (29 inches by 47 inches) is $79 and the price
area of EFGH
b¬(bw)
of the large rug (4 feet 10 inches by 7 feet 10 inches) is $299.
a2
}}2 .
23. What are the areas of the two rugs? What is the ratio of the areas?
b
1363 in.2 and 5452 in.2; 1:4
24. Compare the rug costs. Do you think the large rug is a good buy?
Explain.
FOCUS ON
APPLICATIONS
The price is reasonable; it is slightly less than 4 times the cost of the smaller rug.
TRIANGULAR POOL In Exercises 25–27, use the following information.
The pool at Taliesin West (see page 677) is a right triangle with legs of
length 40 feet and 41 feet.
25. Find the area of the triangular pool, ¤DEF.
820 ft2
26. The walkway bordering the pool is 40 inches
27. Find the area of ¤ABC. What is the area of
the walkway? about 1385.8 ft2; about 565.8 ft2
For Lesson 11.3:
•
Search the Test and Practice
Generator for key words or
specific lessons.
680
RE
L
AL I
FE
For more Mixed Review:
Not drawn
to scale
D
wide, so the scale factor of the similar
triangles is about 1.3:1. Find AB. about 52 ft
ADDITIONAL PRACTICE
AND RETEACHING
• Practice Levels A, B, and C
(Chapter 11 Resource Book,
p. 43)
• Reteaching with Practice
(Chapter 11 Resource Book,
p. 46)
•
See Lesson 11.3 of the
Personal Student Tutor
A
FORT JEFFERSON
is in the Dry Tortugas
National Park 70 miles west
of Key West, Florida. The fort
has been used as a prison, a
navy base, a seaplane port,
and an observation post.
680
28.
B
E
F
¤ABC ~ ¤DEF
FORT JEFFERSON The outer wall of Fort Jefferson, which was
originally constructed in the mid-1800s, is in the shape of a hexagon with
an area of about 466,170 square feet. The length of one side is about
477 feet. The inner courtyard is a similar hexagon with an area of about
446,400 square feet. Calculate the length of a corresponding side in the
inner courtyard to the nearest foot. about 467 ft
Chapter 11 Area of Polygons and Circles
C
Test
Preparation
Æ
Æ
29. e. Since AB and DE are
corresp. sides of ~ ◊,
the ratio of their lengths
is the scale factor. Then
you can write and solve
a proportion;
22.5
15
}} = }}. z = 1
13z º 10
2
29. MULTI-STEP PROBLEM Use the following information about similar
4 ASSESS
triangles ¤ABC and ¤DEF.
The scale factor of ¤ABC to ¤DEF is 15:2.
DAILY HOMEWORK QUIZ
The area of ¤ABC is 25x.
The area of ¤DEF is x º 5.
The perimeter of ¤ABC is 8 + y.
The perimeter of ¤DEF is 3y º 19.
a. Use the scale factor to find the ratio of the area of ¤ABC to the area of
¤DEF. 225:4
Transparency Available
1. The polygons are similar. Find
the ratio (grey to white) of their
perimeters and their areas.
25x
225
}} = }}; 9
xº5
4
b. Write and solve a proportion to find the value of x.
7
5
c. Use the scale factor to find the ratio of the perimeter of ¤ABC to the
perimeter of ¤DEF. 15:2
8+y
15
} = }}; 7
d. Write and solve a proportion to find the value of y. }
3y º 19
2
e. Writing Explain how you could find the value of z if AB = 22.5 and the
Æ
length of the corresponding side DE is 13z º 10. See margin.
★ Challenge
Use the figure shown at the right. PQRS is a parallelogram.
30. Name three pairs of similar triangles and
30. ¤PVQ and ¤RVT, ¤TUS
explain how you know that they are similar.
and ¤TQR, ¤QVR and
¤UVP; all pairs are ~ by
31. The ratio of the area of ¤PVQ to the area of
the AA Similarity Post.
5 25
}} ; }
7 49
2. Determine whether 䉭ACD is
similar to 䉭AEB. If it is, find
the area of 䉭AEB.
C
q
R
B
10
V
12
¤RVT is 9 :25, and the length RV is 10. Find PV. 6
2
EXTRA CHALLENGE
www.mcdougallittell.com
3
P
32. If VT is 15, find VQ, VU, and UT. 9, 5 }5}, 9 }5}
33. Find the ratio of the areas of each pair of
A
S
U
3
¤PVQ and ¤RVT: 9:25; ¤TUS and ¤TQR: 4:25; ¤QVR and ¤UVP: 25:9
3. The price of a small tablecloth
(4 ft by 5 ft) is $45 and the price
of a large tablecloth (96 in. by
120 in.) is $95. Compare the
costs. Is the large tablecloth a
good buy? Explain. The price
MIXED REVIEW
FINDING MEASURES In Exercises 34–37, use the
diagram shown at the right. (Review 10.2 for 11.4)
៣
៣.
36. Find mAC
34. Find mAD . 80°
145°
35. Find m™AEC. 145°
B
E
A
80ⴗ
២
is very reasonable; it is almost
half the cost of 4 times the cost
of the smaller tablecloth.
65ⴗ
37. Find mABC . 215°
C
D
S
38. USING AN INSCRIBED QUADRILATERAL In the
diagram shown at the right, quadrilateral RSTU is
inscribed in circle P. Find the values of x and y, and
use them to find the measures of the angles of RSTU.
R
10x ⴗ
17y ⴗ
19y ⴗ
(Review 10.3)
EXTRA CHALLENGE NOTE
P
8x ⴗ
Challenge problems for
Lesson 11.3 are available in
blackline format in the Chapter 11
Resource Book, p. 51 and at
www.mcdougallittell.com.
T
x = 10, y = 5, m™R = 100°, m™S = 85°, m™T = 80°,
U
m™U = 95°
FINDING ANGLE MEASURES Find the measure of ™1. (Review 10.4 for 11.4)
39. 80°
40. 80°
41. 43°
160ⴗ
1
126ⴗ
50ⴗ
1
110ⴗ
40ⴗ
11.3 Perimeters and Areas of Similar Figures
Additional Test Preparation:
1. Check students’ work. The area should be 9 times the
area of their polygon.
D
†ACD is similar to †AEB.
1
Area of †AEB is 83 }}.
T
similar triangles that you found in Exercise 30.
E
20
1
681
2. Sample answer: Find the ratio of their side lengths. Use
this to find the ratio of their areas. The ratio of their
prices should be proportional to the ratio of their areas.
ADDITIONAL TEST
PREPARATION
1. OPEN ENDED Draw a regular
polygon and find its area. What
is the area of a similar polygon
whose side length is three times
as long as the side length of your
original polygon. See left.
2. WRITING Explain how you can
determine if two products that
are similar polygons are fairly
priced. See left.
681