Similar Polygons

What is a polygon?

A plane figure that has three or more sides and each side intersects exactly two other sides.

Examples: square, triangle, trapezoid…

What are similar polygons?

Two polygons such that their corresponding angles are congruent and the lengths of corresponding sides are proportional.

The symbol for “is similar to” is It is like the congruent symbol without

the equal sign.

Angles and Sides in Similar Polygons

Angles

∠A ≅ ∠ E

∠B ≅ ∠ F

∠C ≅ ∠ G

ΔABC ~ ΔEFG

A

B C

E

F G

Sides

AB ~ EF

AC ~ EG

BC ~ FG

What does that mean?

Corresponding angles are congruent

Lengths of corresponding sides are proportional

∠A =∠E,∠B = ∠F ,

∠C = ∠G ,∠D = ∠H

1

2

2

4==

HEDA

1

2

5.3

7==

GHCD

1

2

3

6==

FGBC

1

2

5.2

5==

EFAB

Thus,

ABCD ~ EFGH

The proportions of the lengths of the corresponding sides of similar polygons are always equal.

The ratio of the lengths of two corresponding sides is called the scale factor.

The scale factor for the previous example is 2:1 or just 2.

1

2

5.2

5==

EFAB

1

2

3

6==

FGBC

1

2

5.3

7==

GHCD

1

2

2

4==

HEDA

How can we know the length of sides in similar figures?

If two figures are similar, one figure is an enlargement of the other. The scale factor tells the amount of enlargement or reduction.

Example 1: If a copy machine is used to copy a drawing or picture, the copy will be similar to the original.

Original Copy

Exact Copy

Copy machine set to 100%

Scale Factor is 1:1

Original Copy

Enlargement

Copy machine is set to 200%

Scale Factor is 1:2

Original Copy

Reduction

Copy machine is set to 50%

Scale Factor is 2:1

Example 1: The triangles CAT and DOG are similar. The larger triangle is an enlargement of the smaller triangle. How long is side GO?

C

A

T G

O

D

1.5 cm

3 cm

2 cm

3 cm

6 cm

? cm

Each side and its enlargement form a pair of sides called corresponding sides.

(1) Corresponding side of TC --> GD

(2) Corresponding side of CA--> DO

(3) Corresponding side of TA--> GO

Length of corresponding sides

GD=3TC=1.5

DO=6CA=3

GO=?TA=2

Ratio of Lengths 3/1.5=2 6/3=2 ?/2=2

The scale factor is 1:2.

Example 2: Quadrangles ABCD and FGCE are similar.

(2) What is scale factor?

(3) What is corresponding side of AD ?

(4) How long is side GC?

(5) What is corresponding side of GC?

(6) How long is side GC? How long is side DC?

3:1

FE

5

BC

GC=5 and DC=12

Poster Design

in. 5

in.

in. 5.3

in. 16 x=

You have been asked to create a poster. 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?

3.5

5

x

16

x⋅=⋅ 5.3516

inches 9.22≈xRemember, mean different things≅≈ ~

One more thing…..

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

If ABCD~EFGH then

HEDA

GHCD

FGBC

EFAB

HEGHFGEFDACDBCAB

====++++++