Similar Figures

Goal 1 Identify Similar Polygons

Goal 2 Find the missing value

Goal 3 Describe a sequence that exhibits the similarity between

two similar two dimensional figures‐

Similar polygons are polygons for which all corresponding angles are congruent and all corresponding sides are proportional.  Example:

Similar PolygonsPolygons are said to be similar if :

a)there exists a one to one correspondence between their sides and angles.

b) the corresponding angles are congruent and

c) their corresponding sides are proportional in lengths.

Definition of Similar Polygons -

Two polygons are SIMILAR if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional.

E

D

C

BA

G

H

I

J

K

In the diagram, pentagon GHIJK is similar to (~) pentagon ABCDE, if all corresponding sides are proportional

GHIJK ~ ABCDE

EA

KG

DE

JK

CD

IJ

BC

HI

AB

GH

Find the value of x, y, and the measure of P if TSV ~ QPR.

x = 6 y = 10.5 P = 86°

Example 1

R

SP

Q

A

B C

D

Example 2Trapezoid 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

4

9

12

12

18

10.5P

Q

R

M

L

N

Example 3

Decide if the triangles are similar.

The triangles are not similar.

Example 4

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

incheswidth

xlength

Width

25.2

5.1

6

4

Sequences That Exhibit Similarity

• You can use sequences of translations, reflections, rotations, and dilations to determine if 2 figures are similar.

• Meaning, look at the 2 figures. If you were to slide, flip, rotate, enlarge, or shrink the first figure, would it then be congruent to the second figure?

/ A / E; / B / F; / C / G; / D / HAB/EF = BC/FG= CD/GH = AD/EH

The scale factor of polygon ABCD to polygon EFGH is 10/20 or 1/2

Scale factor: The ratio of the lengths of two corresponding sides of similar polygons

Scale FactorThe common ratio for pairs of corresponding sides of similar figures

Example:

                        

   

In figure, there are two similar triangles . LMN and PQR.

This ratio is called the scale factor.Perimeter of LMN = 8 + 7 + 10 = 25Perimeter of PQR = 6 + 5.25 + 7.5 = 18.75

33.15.7

10

25.5

7

6

8

Theorem 8.1

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

R

QP

SA

BC

D

SR

AD

QR

CD

PQ

BC

SP

AB

RSQRPQSP

ADCDBCAB

If ABCD ~ SPQR, then

Find the Missing Value

• To find the missing value, set up a proportion, cross multiply, then divide to find the missing value.

Example 5

Parallelogram GIHF is similar to parallelogram LKJF. Find the value of y.

y

24

15

12

F

G

H

I

K

J

L

Y = 19.2

Example 6: 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=3

TC=1.5

DO=6

CA=3

GO=?

TA=2

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

The scale factor is 2.

1.5 cm

2 cm

3 cm

T

C

(1) Each side in the larger triangle is twice the size of the corresponding side in the smaller triangle.

A

G

D

O3 cm

6 cm

? cm

(2) Now, let’s find the length of side GO

i) What side is corresponding side of GO? TA

ii) What is the scale factor? 2

iii) Therefore, GO= scale factor x TA

iv) So, GO= 2 x 2 = 4 cm

What did we just learn about similar polygons ?

Same shapeDifferent size

Corresponding side Size-change factor

Equal angles

Similar polygons

Example 1: Quadrangles ABCD and EFGH are similar. How long is side AD? How long is side GH?

15 cm

? cm

18 cm

12cm7cm

6cm

4cm

?cm

BC

A

D

H

E

GF

(1) What is the scale factor?

(2) What is the corresponding side of AD ?

(3) How long is side AD?

(4) What is the corresponding side of GH?

(5) How long is side GH?

12÷ 4= 3 & 18÷ 6=3

EH

AD = 5

CD

7 x 3 = GH, GH = 21

Example 2 : Figure MORE is similar to Figure SALT. Select the right answer with the one of the given values below.

(1) The length of segment TL is a. 6 cm b. 6.5 cm c. 7 cm d. 7.5 cm

(2) ER corresponds to this segment. a. TS b. TL c. AL d. SA

(3) EM corresponds to this segment. a. TS b. TL c. SA d. AL

(4) The length of segment MO. a. 6 cm b. 6.5 cm c. 7 cm d. 7.5 cm

M O

RE

T

S A

L