similar figures. square limit by m.c. escher escher used a pattern of squares and triangles to...

Similar Figures

Square Limit by M.C. Escher

Escher used a pattern of squares and triangles to create Square Limit.

These two triangles are similar.

Similar figures have the same shape but not necessarily the same size.

Similar Figures

For each part of one similar figure there is a corresponding part on the other figure.Segment AB corresponds to segment DE.

B

A C

D

E

F

Name another pair of corresponding segments.

Similar Figures

Angle A corresponds to angle D.

B

A C

D

E

F

Name another pair of corresponding angles.

•Corresponding sides have lengths that are proportional.

• Corresponding angles are congruent.

Similar Figures

Similar Figures

A D

B C

3 cm

2 cm 2 cm

3 cm

W Z

X Y

9 cm

9 cm

6 cm6 cm

Corresponding sides:

AB corresponds to WX.

AD corresponds to WZ.

CD corresponds to YZ.

BC corresponds to XY.

Similar Figures

A D

B C

3 cm

2 cm 2 cm

3 cm

W Z

X Y

9 cm

9 cm

6 cm6 cm

Corresponding angles:

A corresponds to W. B corresponds to X.

C corresponds to Y.D corresponds to Z.

Similar Figures

A D

B C

3 cm

2 cm 2 cm

3 cm

W Z

X Y

9 cm

9 cm

6 cm6 cm

In the rectangles above, one proportion is

= , or = .AB WX

AD WZ

2 6

3 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.

Missing Measures in Similar Figures

111 y

___ 100 200

____ = Write a proportion using corresponding side lengths.

The cross products are equal.200 • 111 = 100 • y

The two triangles are similar. Find the missing length y and the measure of D.

y is multiplied by 100.22,200 = 100y

22,200 100

______ 100y 100

____ = Divide both sides by 100 to undo the multiplication.

222 mm = y

The two triangles are similar. Find the missing length y.

Angle D is congruent to angle C.

If angle C = 70°, then angle D = 70° .

The two triangles are similar. Find the measure of angle D.

Try This

52 y

___ 50 100

____ = Write a proportion using corresponding side lengths.

The two triangles are similar. Find the missing length y and the measure of B.

5,200 = 50y

5,200 50

_____ 50y 50

___ = Divide both sides by 50 to undo the multiplication.

A B60 m 120 m

50 m 100 m

y52 m

65°

45°

104 m = y

A B60 m 120 m

50 m 100 m

y52 m

65°

45°

Try This

The two triangles are similar. Find the missing length y and the measure of B.

Angle B is congruent to angle A.

If angle A = 65°, then angle B = 65°

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?

ReducedActual

2 54

3w

Using Proportions with Similar Figures

ReducedActual

2 54

3w

54 • 3 = 2 • w

162 = 2w

81 = w

Write a proportion.

The cross products are equal.

w is multiplied by 2.

Divide both sides by 2 to undo the multiplication.

3 cm w cm

2 cm 54 cm

_____ =

Using Proportions with Similar Figures

Try these 5 problems.

These two triangles are similar.

1. Find the missing length x.

2. Find the measure of J.

3. Find the missing length y.

4. Find the measure of P.

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?

36.9°

30 in.

4 in.

90°

12 ft