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Evaluating Algebraic Expressions
5-5 Similar Figures
Warm UpWarm Up
California StandardsCalifornia Standards
Lesson Presentation
PreviewPreview
Evaluating Algebraic Expressions
5-5 Similar Figures
Warm UpSolve each proportion.
b30
39
=1. 5635
y5
=2.
412
p9
=3. 56m
2826
=4.
b = 10 y = 8
p = 3 m = 52
Evaluating Algebraic Expressions
5-5 Similar Figures
Preparation for MG1.2 Construct and read drawings and models made to scale.
California Standards
Evaluating Algebraic Expressions
5-5 Similar Figures
Vocabulary
similar
corresponding sides
corresponding angles
Evaluating Algebraic Expressions
5-5 Similar Figures
Similar figures have the same shape, but not necessarily the same size.
Corresponding sides of two figures are in the same relative position, and corresponding angles 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.
Evaluating Algebraic Expressions
5-5 Similar Figures
43°
43°
Evaluating Algebraic Expressions
5-5 Similar Figures
A is read as “angle A.” ∆ABC is read as “triangle ABC.” “∆ABC ~∆EFG” is read as “triangle ABC is similar to triangle EFG.”
Reading Math
Evaluating Algebraic Expressions
5-5 Similar FiguresAdditional 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.
Evaluating Algebraic Expressions
5-5 Similar FiguresAdditional Example 1 Continued
50 48
The ratios are equal. Rectangle J is similar to rectangle K. The notation J ~ K shows similarity.
The ratios are not equal. Rectangle J is not similar to rectangle L. Therefore, 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.
Evaluating Algebraic Expressions
5-5 Similar FiguresCheck 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.
Evaluating Algebraic Expressions
5-5 Similar FiguresCheck It Out! Example 1 Continued
16 20
The ratios are equal. Rectangle A is similar to rectangle B. The notation A ~ B shows similarity.
The ratios are not equal. Rectangle A is not similar to rectangle C. Therefore, 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.
Evaluating Algebraic Expressions
5-5 Similar Figures
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
Evaluating Algebraic Expressions
5-5 Similar Figures
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?
Evaluating Algebraic Expressions
5-5 Similar FiguresAdditional Example 3: Business Application
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.3 ftx ft
4.5 in.6 in.
=
4.5 • x = 3 • 6 Find the cross products.
side of small triangle base of small triangle
side of large triangle base of large triangle
=
4.5x = 18 Multiply.
Evaluating Algebraic Expressions
5-5 Similar Figures
x = = 4184.5
Solve for x.
Additional Example 3 Continued
The third side of the triangle is 4 ft long.
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?
Evaluating Algebraic Expressions
5-5 Similar FiguresCheck It Out! Example 3
Set up a proportion.24 ftx in.
18 ft4 in.
=
18 ft • x in. = 24 ft • 4 in. 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?side of large triangle side of small triangle
base of large triangle base of small triangle
=
18x = 96 Multiply.
Evaluating Algebraic Expressions
5-5 Similar Figures
x = 5.39618
Solve for x.
Check It Out! Example 3 Continued
The third side of the triangle is about 5.3 in. long.
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?
Evaluating Algebraic Expressions
5-5 Similar FiguresLesson Quiz
Use the properties of similar figures to answer each question.
1. Which rectangles are similar?
A and B are similar.
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. 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.