similar figures 4-3 problem of the day a rectangle that is 10 in. wide and 8 in. long is the same...
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Similar Figures4-3
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.
Similar Figures4-3
Learn to determine whether figures are similar and to find missing dimensions in similar figures.
Similar Figures4-3
Vocabulary
similar
corresponding sides
corresponding angles
Similar Figures4-3
Similar figures have the same shape, but not necessarily the same size.
Corresponding sides and corresponding angles of two figures are in the same relative position.
*Two figures are similar if the lengths of corresponding sides are proportional and the corresponding angles have equal measures.
Similar Figures4-3
Similar Figures4-3
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.
Similar Figures4-3
Additional Example 1 Continued
50 48
The notation J ~ K shows similarity.
The ratios are not equal. Rectangle K is not similar to rectangle L.
20 = 20
length of rectangle Jlength of rectangle K
width of rectangle Jwidth of rectangle K
10 5
42
? =
length of rectangle Jlength of rectangle L
width of rectangle Jwidth of rectangle L
1012
45
?=
Compare the ratios of corresponding sides to see if they are equal.
Similar Figures4-3
Check It Out: Example 1
Which rectangles are similar?
A8 ft
4 ft
B6 ft
3 ft
C5 ft
2 ft
Since the three figures are all rectangles, all the angles are right angles. So the corresponding angles are congruent.
Similar Figures4-3
Check It Out: Example 1 Continued
16 20
A ~ B
Rectangle B is not similar to rectangle C.
24 = 24
length of rectangle Alength of rectangle B
width of rectangle Awidth of rectangle B
8 6
43
? =
length of rectangle Alength of rectangle C
width of rectangle Awidth of rectangle C
8 5
42
?=
Compare the ratios of corresponding sides to see if they are equal.
Similar Figures4-3
14 ∙ 1.5 = w ∙ 10 Find the cross products.
Divide both sides by 10.10w10
2110
=
width of a picturewidth on Web page
height of picture height on Web page
14w
= 101.5
2.1 = w
Set up a proportion. Let w be the width of the picture on the Web page.
The picture on the Web page should be 2.1 in. wide.
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?
Additional Example 2: Finding Missing Measures in Similar Figures
21 = 10w
Similar Figures4-3
56 ∙ 10 = w ∙ 40 Find the cross products.
Divide both sides by 40.40w40
56040
=
width of a paintingwidth of poster
length of painting length of poster
56w
= 4010
14 = w
Set up a proportion. Let w be the width of the painting on the Poster.
The painting displayed on the poster should be 14 in. long.
Check It Out: Example 2
560 = 40w
A painting 40 in. long and 56 in. wide is to be scaled to 10 in. long to be displayed on a poster. How wide should the painting be on the poster for the two pictures to be similar?
Similar Figures4-3
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.6 in.x ft
4.5 in.3 ft
=
4.5 • x = 3 • 6 Find the cross products.
Similar Figures4-3
4.5x = 18
x = = 4184.5
Multiply.
Solve for x.
Additional Example 3 Continued
The third side of the triangle is 4 ft long.
Similar Figures4-3
Check It Out: Example 3
Set up a proportion.24 ftx in.
18 ft4 in.
=
18 • x = 24 • 4 Find the cross products.
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?
Similar Figures4-3
18x = 96
x = 5.39618
Multiply.
Solve for x.
Check It Out: Example 3 Continued
The third side of the triangle is about 5.3 in. long.
Similar Figures4-3
Lesson QuizUse 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.
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. Which rectangles are similar?
17 in.
3.75 ft
A and B are similar.