7.2—Similar Polygons

Identifying Similar PolygonsWhen there is a correspondence between two polygons such that their corresponding angles are congruent and the lengths of corresponding sides are proportional the two polygons are called similar polygons.

ABCD ~ EFGH

HE

DA

GH

CD

FG

BC

EF

AB

means “is similar to”

Writing Similarity StatementsPentagons JKLMN and STUVW are similar. List all the pairs of congruent angles. Write the ratios of the corresponding sides in a statement of proportionality.

Because JKLMN ~ STUVW, you can write J S, K T, L U, M V, and N W.

The statement of proportionality would go as follows:

.WS

NJ

VW

MN

UV

LM

TU

KL

ST

JK

Trapezoid ABCD is similar to trapezoid PQRS. List all the pairs of congruent angles, and write the ratios of the corresponding sides in a statement of proportionality.

A P, B Q, C R, D S

PS

AD

RS

CD

QR

BC

PQ

AB

B C

A D

Q R

P S

Comparing Similar PolygonsDecide whether the figures are similar. If they are similar, write a similarity statement.

To be similar, two polygons must have congruent corresponding angles……and proportionate corresponding sides.

WXYZ ~ PQRS

2

3

10

15

PQ

WX

2

3

4

6

QR

XY

2

3

6

9

RS

YZ

2

3

8

12

SP

ZW

Decide whether the figures are similar. If they are, write the similarity statement.

The triangles are not similar.

M

L

N

P

Q

R

10.5

12

18 12

4

9

Comparing Photographic EnlargementsYou have been asked to create a poster to advertise a field trip to see the Liberty Bell. You have a 3.5 inch by 5 inch photo that you want to enlarge. You want the enlargement to be 16 inches wide. How long will it be?

1. Write a proportion comparing the measurements of the enlargement to the original photo

2. Solve for x

The length will be about 23 inches.

in. 5

in.

in. 3.5

in. 16 x

9.2255.3

16x

You have a photo 4 inches wide by 6 inches long that you want to reduce to fit in a frame that is 1.5 inches wide. How long will the reduced photo be?

2.25 inches

Scale FactorIf two polygons are similar, then the ratio of the lengths of two corresponding sides is called the scale factor.

The rectangular patio around a pool is similar to the pool. Calculate the scale factor of the patio to the pool, and find the ratio of their perimeters.

Since the patio and the pool are similar rectangles, the scale factor is 48 ft : 32 ft or 24 ft : 16 ft, which both simplify to 3 : 2.

The perimeter of the patio is 2(24) + 2(48) = 144 feet and the perimeter of the pool is 2(16) + 2(32) = 96 feet.

Therefore, the ratio of their perimeters is .

2

3or ,

96

144

A painting is similar to the wall on which it is hanging. Calculate the scale factor of the wall to the painting and find the ratio of their perimeters.

2

5,

2

515 ft

6 ft8 ft

3.2 ft

Using Similar Polygons

Theorem If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths.

Quadrilateral JKLM is similar to quadrilateral PQRS. Find the value of z.

1. Write proportion

2. Substitute given values

3. Solve for z60 = 15z4 = z

PQ

JK

QR

KL

z

10

6

15

Parallelogram ABCD is similar to parallelogram GBEF. Find the value of y.

19.2B

E

C

D

A

G

F15

12

24

y

Wrap-Up

How are the perimeters of similar polygons related?The ratio of perimeters is the same as the ratio of

corresponding sides.

The ratio of corresponding sides of two similar hexagons is 3 : 2. What is the ratio of their perimeters? What is the scale factor?3 : 23 : 2