Warm Up

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Warm Up
Problem of the Day
Lesson Presentation
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Warm Up Solve each proportion.
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2.
b 10
y 8
m 52
3.
4.
p 3
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Problem of the Day A rectangle that is 10 in.
wide and 8 in. long is the same shape as one that
is 8 in. wide and x in. long. What is the length
of the smaller rectangle?
6.4 in.
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Learn to determine whether figures are similar,
to use scale factors, and to find missing
dimensions in similar figures.
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Vocabulary
similar congruent angles scale factor
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Similar figures have the same shape, but not
necessarily the same size. Two triangles are
similar if the ratios of the lengths of
corresponding sides are equivalent and the
corresponding angles have equal measures. Angles
that have equal measures are called congruent
angles.
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Additional Example 1 Identifying Similar Figures
Which rectangles are similar?
Since the three figures are all rectangles, all
the angles are right angles. So the corresponding
angles are congruent.
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Additional Example 1 Continued
Compare the ratios of corresponding sides to see
if they are equal.
20 20
The ratios are equal. Rectangle J is similar to
rectangle K. The notation J K shows similarity.
50 ? 48
The ratios are not equal. Rectangle J is not
similar to rectangle L. Therefore, rectangle K is
not similar to rectangle L.
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Check It Out Example 1
Which rectangles are similar?
8 ft
A
B
6 ft
C
5 ft
4 ft
3 ft
2 ft
Since the three figures are all rectangles, all
the angles are right angles. So the corresponding
angles are congruent.
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Check It Out Example 1 Continued
Compare the ratios of corresponding sides to see
if they are equal.
24 24
The ratios are equal. Rectangle A is similar to
rectangle B. The notation A B shows similarity.
16 ? 20
The ratios are not equal. Rectangle A is not
similar to rectangle C. Therefore, rectangle B is
not similar to rectangle C.
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The ratio formed by the corresponding sides is
the scale factor.
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Additional Example 2A Using Scale Factors to
Find Missing Dimensions
A picture 10 in. tall and 14 in. wide is to be
scaled to 1.5 in. tall to be displayed on a Web
page. How wide should the picture be on the Web
page for the two pictures to be similar?
Divide the height of the scaled picture by the
corresponding height of the original picture.
Multiply the width of the original picture by the
scale factor.
14 0.15
2.1
Simplify.
The picture should be 2.1 in. wide.
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Additional Example 2B Using Scale Factors to
Find Missing Dimensions
A toy replica of a tree is 9 inches tall. What is
the length of the tree branch, if a 6 ft tall
tree has a branch that is 60 in. long?
Divide the height of the replica tree by the
height of the 6 ft tree in inches, to find the
scale factor.
Multiply the length of the original tree branch
by the scale factor.
60 .125
7.5
Simplify.
The toy replica tree branch should be 7.5 in long.
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Check It Out Example 2A
A painting 40 in. tall and 56 in. wide is to be
scaled to 10 in. tall to be displayed on a
poster. How wide should the painting be on the
poster for the two pictures to be similar?
Divide the height of the scaled picture by the
corresponding height of the original picture.
Multiply the width of the original picture by the
scale factor.
56 0.25
14
Simplify.
The picture should be 14 in. wide.
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Check It Out Example 2B
A toy replica of a tree is 6 inches tall. What is
the length of the tree branch, if a 8 ft tall
tree has a branch that is 72 in. long?
Divide the height of the replica tree by the
height of the 6 ft tree in inches, to find the
scale factor.
Multiply the length of the original tree branch
by the scale factor.
72 .0625
4.5
Simplify.
The toy replica tree branch should be 4.5 in long.
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Additional Example 3 Using Equivalent Ratios to
Find Missing Dimensions
A T-shirt design includes an isosceles triangle
with side lengths 4.5 in, 4.5 in., and 6 in. An
advertisement shows an enlarged version of the
triangle with two sides that are each 3 ft. long.
What is the length of the third side of the
triangle in the advertisement?
Set up a proportion.
4.5 in. x ft 3 ft 6 in.
Find the cross products.
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Helpful Hint
Draw a diagram to help you visualize the
problems
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Additional Example 3 Continued
Cancel the units.
4.5x 3 6
Multiply.
4.5x 18
Solve for x.
The third side of the triangle is 4 ft long.
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Check It Out Example 3
A flag in the shape of an isosceles triangle with
side lengths 18 ft, 18 ft, and 24 ft is hanging
on a pole outside a campground. A camp t-shirt
shows a smaller version of the triangle with two
sides that are each 4 in. long. What is the
length of the third side of the triangle on the
t-shirt?
Set up a proportion.
18 ft x in. 24 ft 4 in.
Find the cross products.
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Check It Out Example 3 Continued
Cancel the units.
18x 24 4
Multiply.
18x 96
Solve for x.
The third side of the triangle is about 5.3 in.
long.
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Lesson Quiz
Use the properties of similar figures to answer
each question.
1. A rectangular house is 32 ft wide and 68 ft
long. On a blueprint, the width is 8 in. Find
the length on the blueprint.
17 in.
2. Karen enlarged a 3 in. wide by 5 in. tall
photo into a poster. If the poster is 2.25 ft
wide, how tall is it?
3.75 ft
3. Which rectangles are similar?
A and B are similar.