?
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B
A C12 cm
16 cm
8 cm
103°
48° 29°
K
C x
y
L J56 cm
28 cm
12 cm
cm
103°
29°
ESSENTIAL QUESTION
Finding Unknown Measures in Similar TrianglesWhen you measure the height of a door with a measuring tape, you are using
direct measurement. The process of using similar shapes and proportions to find
a measure is called indirect measurement. You can use indirect measurement
to measure things that are difficult to measure directly, like the height of a tree.
▵ABC ~ ▵JKL. Find the unknown measures.
Find the unknown side, x.
AB __
JK = BC
__ KL
8 __
28 = 12
__ x
8 ÷ 4 _____
28 ÷ 4 = 2 _
7
2 _
7 = 12
__ x
x = 42 cm
Find y.
∠K corresponds to ∠B.
y = 103°
Reflect1. Analyze Relationships What other proportion could be used to find
the value of x in the example? Explain.
EXAMPLEXAMPLE 1
A
B
How can you use similar shapes to find unknown measures?
L E S S O N
4.2 Using Similar Shapes
×6
×6
Proportionality—7.5.A Generalize the critical attributes of similarity, including ratios within and between similar shapes.
7.5.A
Write a proportion using corresponding sides. Side AB corresponds to side JK, and side BC corresponds to side KL.Substitute the known lengths of the sides.
Simplify 8 ___ 28 to help you find a factor of 12. 2 is a factor of 12.Since 2 times 6 is 12, multiply 7 times 6 to find the value of x.
Corresponding angles of similar triangles have equal angle measures.
121Lesson 4.2
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B A
J G
H
C
yx
10 cm
6 cm 11.6 cm
5.8 cm
5 cm
59°
31°
J G
H
yx
11.6 cm
5.8 cm
5 cm
s120 cm160 cm
60 cmB A
HC
t
90 cm
m
19°
120°
41°
JG
H
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50 m
?
18 m
9 m
Determining Unknown Measures in Similar Four-Sided ShapesYou can use similar shapes and proportions to find measures of sides
in rectangles and other four-sided shapes.
A volleyball court is a rectangle that is similar in shape to an Olympic-sized
pool. Find the width of the pool.
Let w = the width of the pool.
18 __
50 = 9 __ w
18 __
50 = 9 __ w
w = 25
The pool is 25 meters wide.
EXAMPLE 2
▵ABC ~ ▵JGH. Find the unknown measures.
YOUR TURN
2.
3.
y = x =
s = t =
÷2
÷2
Math TalkMathematical Processes
7.5.A
With similar triangles, you need to find missing sides
and angles. Why do you need to find only missing
sides with rectangles?
Write a proportion using corresponding sides.
Since18 divided by 2 is 9, divide 50 by 2 to find the value of w.
Unit 2122
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54 ft
42 ft ?
36 ft
h
6 ft
1.5 ft4.5 ft
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16 ft
5 ft
4 ft
h
Find the height of the totem pole.
height of person
______________ shadow of person
= height of totem pole
_________________ shadow of totem pole
5 _ 4
= h __ 16
5 _ 4
= h __ 16
h = 20
The totem pole is 20 feet tall.
5. A student who is 4 feet tall stands beside a tree. The tree has a shadow
that is 12 feet long at the same time that the shadow of the student is
6 feet long. Find the height of the tree.
6. A photographer is taking a photo of a statue of
Paul Bunyan, the legendary giant lumberjack.
He measures the length of his shadow and the
shadow cast by the statue. Find the height of the
Paul Bunyan statue.
EXAMPLEXAMPLE 3
YOUR TURN
4. These rectangular
gardens are similar in
shape. Find the width
of the smaller garden.
YOUR TURN
Using Indirect MeasurementWhen you write a proportion to calculate a measurement, you can write ratios
comparing measures of the same object. For example, use the ratio formed by
the height of an object and the length of its shadow.
×4
×4
7.5.A
Write a proportion.
Substitute known values.
Since 4 × 4 = 16, multiply 5 by 4 to find the value of h.
123Lesson 4.2
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B
A C
z
d
8 m 10 m
6 m
Y
X Z
25 m
15 m
37°
53°
B
C
A Z
Y
X
st
8 in.
12 in.
14 in. 42 in.
36 in.35°
85° 60°
B
CA
Z
Y
X x
t
8 in.
28 in.
7 in.
11 in.
44 in.
92°
42° 46°
A
BC
ZY
X
x
a18 ft 18 ft72 ft 72 ft
25 ft
48°
84°
48°
y
x9.5 in. 19 in.
19 in.
19 in. 19 in.
90° 90°
90° 90°
12 cm
6 cm
Q R
T Sf
q
5 cm
A B
D C
Guided Practice
The rectangles in each pair are similar. Find the unknown measures. (Example 2)
5. 6.
▵ABC ~ ▵XYZ in each pair. Find the unknown measures (Example 1)
1. 2.
3. 4.
7. Samantha wants to find the height of a pine tree in her yard. She
measures the height of the mailbox at 3 feet and its shadow at 4.8 feet.
Then she measures the shadow of the tree at 56 feet. How tall is the tree?
(Example 3)
8. Describe how you could use your height and a yardstick to determine the
unknown height of a flagpole.
ESSENTIAL QUESTION CHECK-IN??
Unit 2124
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J
N Z
A
K O
RY
X
ZP Q
ab
20 cm9 cm
8 cm
12 cm
30 cm
40°
89°
R
Y Q
X Z
P
s
y
48 m 64 m40 m30 m
35 m
58°
77°
45°
15 ft 5 ft
3 ft
x
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Name Class Date
Independent Practice4.2
9. A cactus casts a shadow that is 15 ft long.
A gate nearby casts a shadow that is 5 ft
long. Find the height of the cactus.
10. Two ramps modeled with triangles
are similar. Which side of triangle KOA
corresponds to _
JN ? Explain.
11. A building with a height of 14 m casts a
shadow that is 16 m long while a taller
building casts a 24 m long shadow. What is
the height of the taller building?
12. Katie uses a copy machine to enlarge her
rectangular design that is 6 in. wide and
8 in. long. The new width is 10 in. What is
the new length?
13. Art An art exhibit at a local museum
features several similarly shaped metal cubes
welded together to make a sculpture. The
smallest cube has a edge length of 6 inches.
a. What are the edge lengths of the other
cubes if the ratios of similarity to the
smallest cube are 1.25, 4 _ 3 , 1.5, 7 _
4 , and 2
respectively?
b. If the artist wanted to add a smaller
cube with an edge length with a ratio
of 2 _ 3 to the sculpture, what size would
the cube be?
c. Why do you only have to find the
length of one edge for each cube?
▵XYZ ~ ▵PQR in each pair. Find the unknown
measures.
14.
15.
7.5.A
125Lesson 4.2
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Work Area
16. Two common envelope sizes are 3 1 _ 2 in. × 6 1 _
2 in. and 4 in. × 9 1 _
2 in. Are these
envelopes similar? Explain.
17. A pair of rectangular baking pans come in a set together for $15. One pan
is 13 inches by 9 inches and the other pan is 6 inches by 6 inches. Without
doing any calculations, how can you tell that these pans are not similar?
18. Draw Conclusions In the similar triangles used in indirect measurement
with the shadows of a flagpole and a person, which sides of the triangles
represent the rays of the sun?
19. Make a Conjecture Do you think it is possible to use indirect
measurement with shadows if the sun is directly overhead? Explain.
20. Analyze Relationships Joseph's parents have planted two gardens. One
is square and has an area of 25 ft 2 . The other one has two sides equal to 2 _ 3
of one side of the square, and the other two sides equal to 5 _ 2 of one side
of the square.
a. Find the dimensions of the other garden. Explain how you found
your answer.
b. Find the area of the other garden.
FOCUS ON HIGHER ORDER THINKING
Unit 2126
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