Similar Polygons

Informal Definition of Similar Figures

Two figures are similar if they have the same

shape.(They do not necessarily have the

same size.)

Formal Definition of Similar Polygons

Two polygons are similar if their corresponding angles are congruent,and(lengths of) corresponding sides are proportional.

C

A

B

E

DF1. A D

2. B E

3. C FAB

DEAC

DF

BC

EF

Proportional(Lengths of) sides are proportional iff ratios of (lengths of) corresponding sides are equal.For example:

so the sides are proportional.

5 9

7

18

10

14

C

A

B

Z

YX

18

914

7

10

5 2

The ratio of corresponding sides of similar polygons.

Example The scale factor from ABC to _____

is____.from ZYX to ABC

is____.

Scale Factor

7 9

5

18

10

14

C

A

B

Z

YX

ZYX1/2

2

Naming Similar Polygons

**Must match the corresponding letters**

S

Q

D

C

B

AR

P

ABDC ~ RPSQ

Applying the Definition - Angles

S

Q

D

C

B

AR

P

B P D SA R C Q

**Must match the corresponding vertices**

Applying the Definition - Sides

S

Q

D

C

B

AR

P**Must match the corresponding

sides**

BD

PS

AC

RQ

AB

RP

DC

SQ

Proportional means all of

the ratios are equal!

Example 1

S

Q

D

C

B

AR

P

18

14

28

248

Given

Find the lengths of the missing sides.

ABDC ~ RPSQ

AC AB

RQ

Example 1

S

Q

D

C

B

AR

P

18

14

28

248

Given

Find AC

Step 1: Write out a proportion of for the sides.

(Be sure to match up corresponding letters!)

ABDC ~ RPSQ

RP

AB8

RP

Example 1

S

Q

D

C

B

AR

P

18

14

28

248

Given

Find the lengths of

the missing sides.

Step 2: Replace the sides with the lengths from the problem.

ABDC ~ RPSQ

AC

RQ

14 18

Example 1

S

Q

D

C

B

AR

P

18

14

28

248

Given

Find the lengths of

the missing sides.

Step 3: Cross-multiply and solve.

ABDC ~ RPSQ

14

AC 8

1818 14 8AC

14 8

18AC

56

9AC

Example 1

S

Q

D

C

B

AR

P

18

14

28

248

Given ABDC ~ RPSQ

You should be able to

find CD and BD as well!

Example 2

S

Q

D

C

B

AR

P

18

14

28

248

Given

Find the scale factor of ABDC to RPSQ

ABDC ~ RPSQ

BD

28

AC

14

8

18

DC

24

Remember the scale factor is same as the ratio of the sides.

Always put the first polygon mentioned in the numerator.

BD

PS

AC

RQ

AB

RP

DC

SQ

The scale factor of ABDC to RPSQ is 4

9

Example 3

ABCD EFGH. Solve for x, y and z.

EH EF

AD AB

5

z

x

10y

15

30

20

B C

AD

F G

E

H

10

15 20

x 20

5 10

y

30 20

10z

x = 7.5

DC AB

HG EF

FG EF

BC AB

Step 1: Write a proportion using names of sides.Step 2: Substitute values.Step 3: Cross-multiply to solve.Step 4: Repeat to find other values.

y = 10 z = 15

Dilations and Scale FactorA dilation is a transformation that changes the size of an object.

The scale factor is the ratio of the lengths of the corresponding sides of two similar polygons. It indicates the relative size of one polygon compared the another.

Congruent Triangles

Are congruent triangles similar?

What is the scale factor between two congruent triangles?