Transcript Ch 5-5 Similar Figures

5-5 Similar Figures
California
Standards
Preparation for MG1.2 Construct and
read drawings and models made to scale.
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.
5-5 Similar Figures
43°
43°
5-5 Similar Figures
Reading Math
A is read as “angle A.” ∆ABC is read as “triangle
ABC.” “∆ABC ~∆EFG” is read as “triangle ABC is
similar to triangle EFG.”
5-5 Similar Figures
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.
5-5 Similar Figures
Additional Example 1 Continued
Compare the ratios of corresponding sides to see if they are equal.
length of rectangle J
length of rectangle K
10 ? 4
5 =2
20 = 20
width of rectangle J
width of rectangle K
The ratios are equal. Rectangle J is similar to
rectangle K. The notation J ~ K shows similarity.
length of rectangle J
length of rectangle L
10 ? 4
width of rectangle J
12 = 5
width of rectangle L
50  48
The ratios are not equal. Rectangle J is not
similar to rectangle L. Therefore, rectangle K is
not similar to rectangle L.
5-5 Similar Figures
Check It Out! Example 1
Which rectangles are similar?
8 ft
A
4 ft
6 ft
B
3 ft
5 ft
C
2 ft
Since the three figures are all rectangles, all the
angles are right angles. So the corresponding
angles are congruent.
5-5 Similar Figures
Check It Out! Example 1 Continued
Compare the ratios of corresponding sides to see if they are equal.
length of rectangle A
length of rectangle B
8 ? 4
width of rectangle A
6 =3
width of rectangle B
24 = 24
The ratios are equal. Rectangle A is similar to
rectangle B. The notation A ~ B shows similarity.
length of rectangle A
length of rectangle C
8 ? 4
width of rectangle A
5= 2
width of rectangle C
16  20
The ratios are not equal. Rectangle A is not similar
to rectangle C. Therefore, rectangle B is not similar
to rectangle C.
5-5 Similar Figures
Additional Example 2: Finding Missing Measures in
Similar Figures
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?
Set up a proportion. Let w be the width of the picture on the
Web page.
14
10
width of a picture
height of picture
=
1.5
width on Web page w
height on Web page
14 ∙ 1.5 = w ∙ 10
Find the cross products.
21 = 10w
21
10w
Divide both sides by 10.
=
10
10
2.1 = w
The picture on the Web page should be 2.1 in. wide.
5-5 Similar Figures
Check It Out! Example 2
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?
Set up a proportion. Let w be the width of the painting on the
Poster.
width of a painting
width of poster
56 ∙ 10 = w ∙ 40
56
40
=
w
10
length of painting
length of poster
Find the cross products.
560 = 40w
560
40w
Divide both sides by 40.
=
40
40
14 = w
The painting displayed on the poster should be 14 in. long.
5-5 Similar Figures
Additional 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?
side of small triangle
base of small triangle
4.5 in. = 3 ft
6 in.
x ft
4.5 • x = 3 • 6
4.5x = 18
=
side of large triangle
base of large triangle
Set up a proportion.
Find the cross products.
Multiply.
5-5 Similar Figures
Additional Example 3 Continued
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?
x = 18 = 4
4.5
Solve for x.
The third side of the triangle is 4 ft long.
5-5 Similar Figures
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?
side of large triangle
side of small triangle
18 ft = 24 ft
4 in.
x in.
=
base of large triangle
base of small triangle
Set up a proportion.
18 ft • x in. = 24 ft • 4 in. Find the cross products.
18x = 96
Multiply.
5-5 Similar Figures
Check It Out! Example 3 Continued
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?
x = 96  5.3
18
Solve for x.
The third side of the triangle is about 5.3 in. long.