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8-4 Similar Figures
Warm Up
Problem of the Day
Lesson Presentation
Course 1
8-4 Similar Figures
Warm Up
Fill in the missing value.
1.
c = 2 qt
2. 180 in. =
3. 3 tons =
4.
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8
yd
lb
min = 2,760 s
5
6,000
45
8-4 Similar Figures
Problem of the Day
How many 8 in. by 10 in. rectangular
tiles would be needed to cover a 16 ft
by 20 ft floor?
576
Course 1
8-4 Similar Figures
Learn to use ratios to identify similar
figures.
Course 1
8-4 Similar
Insert Lesson
FiguresTitle Here
Vocabulary
similar
corresponding sides
corresponding angles
Course 1
8-4 Similar Figures
Two or more figures are similar if
they have exactly the same shape.
Similar figures may be different
sizes.
Similar figures have corresponding
sides and corresponding angles.
• Corresponding sides have
lengths that are proportional.
• Corresponding angles are
congruent.
Course 1
8-4 Similar Figures
A
3 cm
2 cm
B
D
2 cm
3 cm
W
9 cm
6 cm
Z
6 cm
C
Corresponding sides:
X
9 cm
Y
Corresponding angles:
AB corresponds to WX.
BC corresponds to XY.
A corresponds to
B corresponds to
W.
X.
CD corresponds to YZ.
AD corresponds to WZ.
C corresponds to
D corresponds to
Y.
Z.
Course 1
8-4 Similar Figures
A
3 cm
2 cm
B
D
2 cm
3 cm
W
9 cm
6 cm
Z
6 cm
C
X
9 cm
Y
In the rectangles above, one proportion is
AB
AD
2
3
=
, or
= .
WX WZ
6
9
If you cannot use corresponding side lengths to
write a proportion, or if corresponding angles are
not congruent, then the figures are not similar.
Course 1
8-4 Similar Figures
Additional Example 1: Finding Missing
Measures in Similar Figures
The two triangles are similar. Find the
missing length y and the measure of D.
100
111
Write a proportion using
____
= ___
200
y
corresponding side lengths.
200 • 111 = 100 • y The cross products are equal.
Course 1
8-4 Similar Figures
Additional Example 1 Continued
The two triangles are similar. Find the
missing length y and the measure of D.
22,200 = 100y
22,200 = ____
100y
______
100
100
y is multiplied by 100.
Divide both sides by 100
to undo the multiplication.
222 mm = y
Angle D is congruent to angle C, and m
m
Course 1
D = 70°
C = 70°.
8-4 Similar Figures
Try This: Example 1
The two triangles are similar. Find the
missing length y and the measure of B.
A 60 m
65°
50 m
45° 52 m
B
120 m
100 m
y
50
52
____
___
=
100
y
100 • 52 = 50 • y
Course 1
Write a proportion using
corresponding side lengths.
The cross products are equal.
8-4 Similar Figures
Try This: Example 1 Continued
The two triangles are similar. Find the
missing length y and the measure of B.
5,200 = 50y
5,200 = 50y
_____
___
50
50
104 m = y
y is multiplied by 50.
Divide both sides by 50 to
undo the multiplication.
Angle B is congruent to angle A, and m
65°.
m
Course 1
B = 65°
A=
8-4 Similar Figures
Additional Example 2: Problem Solving Application
This reduction is similar to a
picture that Katie painted. The
height of the actual painting is
54 centimeters. What is the
width of the actual painting?
1
Understand the Problem
The answer will be the width of the actual painting.
List the important information:
• The actual painting and the reduction above are similar.
• The reduced painting is 2 cm tall and 3 cm wide.
• The actual painting is 54 cm tall.
Course 1
8-4 Similar Figures
Additional Example 2 Continued
2
Make a Plan
Draw a diagram to represent the situation.
Use the corresponding sides to write a
proportion.
Actual
Reduced
2
54
3
w
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8-4 Similar Figures
Additional Example 2 Continued
3
Solve
2 cm
3 cm
_____
=
w cm Write a proportion.
54 cm
54 • 3 = 2 • w The cross products are equal.
162 = 2w
162
2w
____
___
=
2
2
w is multiplied by 2.
Divide both sides by 2 to
undo the multiplication.
81 = w
The width of the actual painting is 81 cm.
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8-4 Similar Figures
Additional Example 2 Continued
4
Course 1
Look Back
Estimate to check your answer. The ratio of
the heights is about 2:50 or 1:25. The ratio
of the widths is about 3:90, or 1:30. Since
these ratios are close to each other, 81 cm
is a reasonable answer.
8-4 Similar Figures
Try This: Example 2
This reduction is similar to a
picture that Marty painted. The
height of the actual painting is
39 inches. What is the width of
the actual painting?
1
4 in.
3 in.
Understand the Problem
The answer will be the width of the actual painting.
List the important information:
• The actual painting and the reduction above are similar.
• The reduced painting is 3 in. tall and 4 in. wide.
• The actual painting is 39 in. tall.
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8-4 Similar Figures
Try This: Example 2 Continued
2
Make a Plan
Draw a diagram to represent the situation.
Use the corresponding sides to write a
proportion.
Actual
Reduced
3
39
4
w
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8-4 Similar Figures
Try This: Example 2 Continued
3
Solve
3 in
4 in
_____
= ____ Write a proportion.
39 in
w in
39 • 4 = 3 • w The cross products are equal.
156 = 3w
156 = ___
3w
____
3
3
w is multiplied by 3.
Divide both sides by 3 to
undo the multiplication.
52 = w
The width of the actual painting is 52 inches.
Course 1
8-4 Similar Figures
Try This: Example 2 Continued
4
Course 1
Look Back
Estimate to check your answer. The ratio of
the heights is about 4:40, or 1:10. The
ratio of the widths is about 5:50, or 1:10.
Since these ratios are the same, 52 inches
is a reasonable answer.
8-4 Similar
Insert Lesson
FiguresTitle Here
Lesson Quiz
These two triangles are similar.
1. Find the missing length x. 30 in.
2. Find the measure of
J. 36.9°
3. Find the missing length y. 4 in.
4. Find the measure of
P.
90°
5. Susan is making a wood deck from plans for
an 8 ft by 10 ft deck. However, she is going to
increase its size proportionally. If the length is
to be 15 ft, what will the width be?
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12 ft