Learn to determine whether figures are similar, to use scale factors, and to find missing dimensions

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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
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The heights of letters in newspapers and on
billboards are measured using points and picas.
There are 12 points in 1 pica and 6 picas in one
inch.
A letter 36 inches tall on a billboard would be
216 picas, or 2592 points. The first letter in
this paragraph is 12 points.
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Congruent figures have the same size and shape.
Similar figures have the same shape, but not
necessarily the same size. The As in the table
are similar. They have the same shape, but they
are not the same size.
The ratio formed by the corresponding sides is
the scale factor.
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Additional Example 1 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?
To find the scale factor, divide the known
measurement of the scaled picture by the
corresponding measurement of the original picture.
0.15
Then multiply the width of the original picture
by the scale factor.
2.1
14 0.15 2.1
The picture should be 2.1 in. wide.
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Try This Example 1
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?
To find the scale factor, divide the known
measurement of the scaled painting by the
corresponding measurement of the original
painting.
0.25
Then multiply the width of the original painting
by the scale factor.
14
56 0.25 14
The painting should be 14 in. wide.
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Additional Example 2 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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Additional Example 2 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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Try This Example 2
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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Try This Example 2 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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Remember!
The following are matching, or
corresponding ?A and ?X
A
X
Y
Z
C
B
?B and ?Y
?C and ?Z
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Additional Example 3 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 3 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.
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Try This Example 3
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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Try This Example 3
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.
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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.