7.2 Similar Polygons

Geometry

Mr. Peebles

Spring 2013

Geometry Bell Ringer

• The rectangular patio

around a pool is

similar to the pool as

shown. Calculate the

scale factor of the

patio to the pool, and

find the ratio of their

perimeters.

• The ratio is the

• Patio/Pool

18 ft 27 ft

28 ft

42 ft

Geometry Bell Ringer

• The rectangular patio

around a pool is

similar to the pool as

shown. Calculate the

scale factor of the

patio to the pool, and

find the ratio of their

perimeters.

• The ratio is the

• Answer: 3/2 or 1.5

is the Scale Factor.

18 ft 27 ft

28 ft

42 ft

Bell Ringer: Solve For X

4 x+10

8 x

=

Bell Ringer: Solve For X

4 x+10

8 x

=

Answer: x = -20

Daily Learning Target (DLT)

Tuesday February 26, 2013 • “I can understand, apply, and remember

how to identify similar polygons notably

in real-life problems.”

Identifying 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.

• Remember that geometric figures that are similar have the same shape and angles but have different sizes.

• In the next slide, ABCD is similar to EFGH. The symbol ~ is used to indicate similarity. So, ABCD ~ EFGH.

Similar polygons

B

C

AD

F

G

E H

AB = =

EF

BC

FG =

CD

GH

DA

HE

AB =

EF

Polygon ABCD

Polygon EFGH

Ex. 1: Writing Similarity Statements

• Pentagons JKLMN and STUVW are

similar. List all the pairs of congruent

angles. Write the ratios of the

corresponding sides in a statement of

proportionality. J K

L

M

N

S T

U

V

W

Ex. 1: Writing Similarity Statements

J K

L

M

N

S T

U

V

W

Because JKLMN ~ STUVW, you can

write J S, K T, L U, M

V AND N W.

You can write the

proportionality statement

as follows:

KL

TU =

JK =

ST

MN

VW =

LM =

UV

NJ

WS

Ex. 2: Comparing Similar Polygons

• Decide whether the figures are similar. If

they are similar, write a similarity

statement.

15

12

9

6X

W

Z

Y

10

8

6

4Q

P

S

R

15

12

9

6X

W

Z

Y

10

8

6

4Q

P

S

RSOLUTION:

As shown, the corresponding

angles of WXYZ and PQRS

are congruent. Also, the

corresponding side lengths

are proportional.

WX

PQ =

15

10 =

3

2

XY

QR =

6

4 =

3

2

YZ

RS =

9

6 =

3

2

WX

PQ =

15

10 =

3

2 So, the two figures are

similar and you can write

WXYZ ~ PQRS.

Ex. 3: Comparing Photographic

Enlargements

• POSTER DESIGN. You 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? Hint… you will need to

set up a proportion comparing the length

and width of the picture and frame.

Solution:

• To find the length of the enlargement,

you can compare the enlargement to the

original measurements of the photo.

16 in.

3.5 in. =

x in.

5 in.

x = 16

3.5 x∙ 5

x ≈ 22.9 inches

The length of the

enlargement will be about 23

inches.

5

3.5

Trip to Liberty Bell

March 24th,

Sign up

today!

Theorem 8.1

• Theorem 8.1: If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding parts.

• If KLMN ~ PQRS, then

P

S

Q

RK

N

L

M

KL + LM + MN + NK

PQ + QR + RS + SP =

KL

PQ

LM

QR

MN

RS

NK

SP = = =

Using similar polygons in real life

• If two polygons are similar, then the ratio

of lengths of two corresponding sides is

called the scale factor. In Example 2 on

the previous page, the common ratio of

is the scale factor of WXYZ to PQRS.

3

2

Ex. 4: Using similar polygons

• The rectangular patio

around a pool is

similar to the pool as

shown. Calculate the

scale factor of the

patio to the pool, and

find the ratio of their

perimeters.

• The ratio is the

• Patio/Pool

16 ft 24 ft

32 ft

48 ft

• Because the rectangles are

similar, the scale factor of

the patio to the pool is 48

ft: 32 ft. , which is 3:2 in

simplified form.

• 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 The ratio of the

perimeters is

16 ft 24 ft

32 ft

48 ft 144

96

3

2 , or

NOTE:

• Notice in Example 4 that the ratio of

perimeters is the same as the scale factor

of the rectangles. This observation is

generalized in the following theorem.

Ex. 5: Using Similar Polygons

• Quadrilateral JKLM is

similar to PQRS. Find the

value of z.

Set up a proportion that contains PQ

15

10J

M L

K

Z

6

S

PQ

R

KL

QR

JK

PQ =

Write the proportion.

Ex. 5: Using Similar Polygons

• Quadrilateral JKLM is

similar to PQRS. Find the

value of z.

Set up a proportion that contains PQ

15

10J

M L

K

Z

6

S

PQ

R

KL

QR

JK

PQ =

15

6

10

Z =

Z = 4

Write the proportion.

Substitute

Cross multiply and divide by 15.

Assignment

Pages 376 (1-16)

Assignment

Pages 376 (1-16)

1. Angle JHY 10. ABCD ~ FGHE

2. Angle R 11. They are NOT Similar

3. Angle JXY 12. ∆ABC ≅ ∆FED

4. Side HY 13. x = 4, y = 3

5. Side JT 14. x = 20, y = 17.5, z = 7.5

6. Side HY 15. x = 16, y = 4.5, z = 7.5

7. They are NOT Similar 16. x = 6

8. QRST ~ XWZY y = 8

9. KLMJ ~ PQNO z = 10

Today’s Assignment

Pages 376-378 (21-29, 31-39, 48, 53-56)

Geometry Exit Quiz – 5 Points

• The rectangular patio

around a pool is

similar to the pool as

shown. Calculate the

scale factor of the

patio to the pool, and

find the ratio of their

perimeters.

27 ft 36 ft

39 ft

52 ft