ratios and similar polygons - learning resource center

Ratios and Similar Polygons

A useful way to compare two quantities is write a ratio. If your family has 2 dogs and 3 cats, the ratio of dogs to cats is 2/3 or 2:3.

Must be

the same

units!!

An equation that equates two

ratios is a proportion.

Properties of Proportion

Properties of Proportion

Solving Proportions

Solving Proportions

Solution: 4x = 5*8x = 10

Solve the Proportion

Solve the Proportion

Solution: 4(x+5) = 16(x+2)4x+20 = 16x + 32

-12 = 12 xx = -1

AB

CD

CF

AB

AB

CD

CF

AB

Solution:

2 or 14 2

92

Solve for x

Solve for x

Solution: 2 = x5 20

2*20 = 5x

8 = x

The Area of a rectangle is 108. The

ratio of length to width is 3:4.

What is the length and width of the

rectangle?

l

w

The Area of a rectangle is 108. The

ratio of length to width is 3:4.

What is the length and width of the

rectangle?

l

wSolution:

3x*4x = 10812x^2 = 108

x^2 = 9 l = 3x = 3*3 = 9x = 3 w = 4x = 4*3 = 12

The measures of the angles in a

triangle are in the extended ratio

1:4:7. Find the measure of each

angle.

The measures of the angles in a triangle are in

the extended ratio 1:4:7. Find the measure of

each angle.

Solution: 1x+4x+7x = 180

12x = 108x = 9

x = 9o 4x = 36o 7x = 63o

Similar Polygons

• When 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

Quad ABCD ~Quad EFGH

• ~ means Similar

• List all the the pairs of congruent angles.

• Write the ratios of the corresponding sides in a statement of proportionality

Quad ABCD ~Quad EFGH• ~ means Similar

• List all the the pairs of congruent angles.

<A~ <E <B~ <F <D~ <H <C~ <G

• Write the ratios of the corresponding sides in a statement of proportionality

AB = BC = CD = AD

EF FG GH EH

ABC DEF • List all congruent

angles.

• Write the ratios of corresponding sides.

ABC DEF • List all congruent angles.

<A~ <D <B~ <E <C~ <F

• Write the ratios of corresponding sides.

AB = BC = AC

DE EF DF

Theorem

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

Definition

If two polygons are similar, then the ratio of the lengths of two corresponding sides is called the Scale Factor

Pentagon ABCDE ~ JKLMN

• What is the scale factor?

• What is the value of x?

Pentagon ABCDE ~ JKLMN

• What is the scale factor?

5/6

• What is the value of x?

5 = 4

6 x

5x = 4*6

x = 24/5 = 4.8

Are these Triangles similar?

• If so, write a similarity statement.

Are these Triangles similar?

• If so, write a similarity statement.

Solution:

2 = 3 = 4

4 6 8

Yes

ABC ~ XYZ

Decide whether the figures are similar.

If they are, write the similarity statement.

Decide whether the figures are similar.

If they are, write the similarity statement.

Solution:

4 = 9 = 12

10.5 12 18

NO

You have a photo 4 in. wide by 6 in. long

that you want to reduce to fit in a frame

that is 1.5 in. wide. How long will the

reduced photo be?

4 in

1.5 in

6 in

x

You have a photo 4 in. wide by 6 in. long that

you want to reduce to fit in a frame that is 1.5

in. wide. How long will the reduced photo be?

Solution: 4 = 1.56 l 4l = 9 l = 2.25 in.

4 in

1.5 in

6 in

x

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.

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.

Solution:

3.2 = 6 = 2 Ratio of perimeters is the same.8 15 5

Parallelogram ABCD is similar to

parallelogram GBEF.

Find the value of y.

B

E

C

F

D

G A

15

24

12

y

Parallelogram ABCD is similar to

parallelogram GBEF. Find the value of y.

B

E

C

F

D

G A

15

24

12

y

Solution:

12 = 15y 24 12 * 24 = 15y y = 19.2

ΔUVW is similar to

ΔYXW.

Find the value of a.

ΔUVW is similar to

ΔYXW.

Find the value of a.

Solution:

12 = 6.4a 5 12 * 5 = 6.4a a = 9.4

Solution:

5 = 12 5(x – 1) = 10 *1210 x - 1 5x – 5 = 120 x = 25

5 = 13 5(y + 20) = 10 *1310 y + 20 5y + 100 = 130 y = 6

Sometimes, Always or Never?

• Two isosceles triangles are similar

• Isosceles trapezoids are similar

• Squares are similar

• Equilateral triangles are similar

• An isosceles triangle and a scalene triangle are similar.

Sometimes, Always or Never?

• Two isosceles triangles are similar

Sometimes

• Isosceles trapezoids are similar

Sometimes

• Squares are similar

Always

• Equilateral triangles are similar

Always

• An isosceles triangle and a scalene triangle are similar.

Never