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Similar Figure
Word Problems
8-5 C1: Learn to use proportions and
similar figures to find unknown measures.
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Vocabulary
indirect measurement
One way to find a height that you cannot
measure directly is to use similar figures
and proportions. This method is called
indirect measurement.
Additional Example 1: Using Indirect Measurement
Try This: Example 1
Use the similar triangles to find the height of the
tree.
6
2
Write a proportion using corresponding
__
= __
sides.
h
7
Use the similar triangles to find the height of the
tree.
Write a proportion using corresponding
6
3
__
= __
sides.
h
9
h • 2 = 6 • 7 The cross products are equal.
h • 3 = 6 • 9 The cross products are equal.
2h = 42
h is multiplied by 2.
3h = 54
h is multiplied by 3.
6 ft.
2h = ___
42 Divide both sides
___
by 2 to undo
2
2
multiplication.
h = 21
The tree is 21 feet tall.
3h = ___
54
___
3
3
h = 18
Divide both sides
by 3 to undo
multiplication.
h
3 ft.
9 ft.
The tree is 18 feet tall.
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Additional Example 2: Measurement Application
A rocket casts a shadow that is 91.5 feet long.
A 4-foot model rocket casts a shadow that is 3
feet long. How tall is the rocket?
h
91.5 Write a proportion using
__
= ____ corresponding sides.
4
3
4 • 91.5 = h • 3 The cross products are equal.
366 = 3h
h is multiplied by 3.
3h Divide both sides by
366 = ___
___
3 to undo
3
3
multiplication.
122 = h
The rocket is 122 feet tall.
Try This: Example 2
A building casts a shadow that is 72.5 feet long
when a 4-foot model building casts a shadow
that is 2 feet long. How tall is the building?
h
72.5 Write a proportion using
__
= ____
corresponding sides.
4
2
4 • 72.5 = h • 2 The cross products are equal.
290 = 2h
h is multiplied by 2.
2h Divide both
290 = ___
___
sides by 2 to
2
2
undo
145 = h
multiplication.
The building is 145 feet tall.
h
4 ft
72.5 ft
2 ft
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Lesson Quiz
1. On a sunny day, a telephone pole casts a
shadow 21 ft long. A 5-foot-tall mailbox next
to the pole casts a shadow 3 ft long. How tall
is the pole? 35 feet
8-6 C1: Learn to read and use map scales
and scale drawings.
2. On a sunny afternoon, a goalpost casts a 75 ft
shadow. A 6.5 ft football player next to the
goal post has a shadow 19.5 ft long. How tall
is the goalpost? 25 feet
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Vocabulary
scale drawing
scale
The map shown is a scale
drawing. A scale drawing is
a drawing of a real object
that is proportionally smaller
or larger than the real object.
In other words,
measurements on a scale
drawing are in proportion to
the measurements of the real
object.
A scale is a ratio between two sets of
measurements. In the map above, the scale is 1
in:100 mi. This ratio means that 1 inch on the map
represents 100 miles.
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Additional Example 1: Finding Actual Distances
The scale on a map is 4 in: 1 mi. On the map,
the distance between two towns is 20 in.
What is the actual distance?
Write a proportion using the scale.
4 in.
20 in.
____
= _____ Let x be the actual number of
1 mi
x mi
miles between the two towns.
1 • 20 = 4 • x
The cross products are equal.
20 = 4x
20
4x
___
= ___
4
4
5=x
Helpful Hint
In Additional Example 1, think “4 inches is 1
mile, so 20 inches is how many miles?” This
approach will help you set up proportions in
similar problems.
x is multiplied by 4.
Divide both sides by 4 to undo
multiplication.
The actual distance between the two towns is 5 miles.
Try This: Example 1
The scale on a map is 3 in: 1 mi. On the
map, the distance between two cities is 18
in. What is the actual distance?
Write a proportion using the scale.
3 in.
18 in.
____
= _____ Let x be the actual number of
1 mi
x mi
miles between the two cities.
1 • 18 = 3 • x
The cross products are equal.
18 = 3x
18
3x
___
= ___
3
3
6=x
x is multiplied by 3.
Divide both sides by 3 to undo
multiplication.
The actual distance between the two cities is 6 miles.
Additional Example 2B: Astronomy Application
B. The actual distance from Earth to Mars is
about 78 million kilometers. How far apart
should Earth and Mars be drawn?
Write a proportion. Let x be
1 in.
x in.
___________
= __________ the distance from Earth to
30 million km 78 million km
30 • x = 1 • 78
30x = 78
Mars on the drawing.
The cross products are equal.
x is multiplied by 30.
Divide both sides by 30
30x = ___
78
___
to undo multiplication.
30
30
3
__
x=25
3 inches apart.
Earth and Mars should be drawn 2__
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Additional Example 2A: Astronomy Application
A. If a drawing of the planets were made using
the scale 1 in:30 million km, the distance from
Mars to Jupiter on the drawing would be about
18.3 in. What is the actual distance between
Mars to Jupiter?
Write a proportion. Let x
1 in.
18.3 in.
___________
= _________ be the actual distance
30 million km x million km from Mars to Jupiter.
30 • 18.3 = 1 • x
The cross products
are equal.
549 = x
The actual distance from Mars to Jupiter is about 549
million km.
Try This: Additional Example 2A
A. If a drawing of the planets were made using
the scale 1 in:15 million km, the distance from
Mars to Venus on the drawing would be about 8
in. What is the actual distance from Mars to
Venus?
Write a proportion. Let x
1 in.
8 in.
___________
= _________ be the distance from
15 million km x million km Mars to Venus.
15 • 8 = 1 • x
The cross products
are equal.
120 = x
The actual distance from Mars to Venus is about 120
million km.
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Insert Lesson Title Here
Try This: Example 2B
B. The distance from Earth to the Sun is about
150 million kilometers. How far apart should
Earth and the Sun be drawn?
Write a proportion. Let x be
1 in.
x in.
________
= ________ the distance from Earth to
15 mil km
150 mil km
the Sun on the drawing.
15 • x = 1 • 150
The cross products are equal.
15x = 150
x is multiplied by 15.
15x
150
___
Divide both sides by 15 to
= ____
15
15
undo multiplication.
x = 10
Lesson Quiz
On a map of the Great Lakes, 2 cm = 45 km.
Find the actual distance of the following,
given their distances on the map.
1. Detroit to Cleveland = 12 cm 270 km
2. Duluth to Nipigon = 20 cm 450 km
3. Buffalo to Syracuse = 10 cm 225 km
4. Sault Ste. Marie to Toronto = 33 cm 742.5 km
Earth and the Sun should be drawn 10 inches apart.
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