Bell Ringer 6.

7.

Similar Polygons

Two polygons are similar polygons if corresponding angles are congruent and corresponding side length are proportional.

Example 1 Use Similarity StatementsPRQ ~ STU.

List all pairs of congruent angles.a.

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

b.

Check that the ratios of corresponding sides are equal.

c.

SOLUTION

P S, R T, and Q U.a.

PRST

RQTU

QPUS

= =b.

c.PRST

108

54= = , RQ

TU2016

54= = , and QP

US1512

54= = .

The ratios of corresponding sides are all equal to 5

4.

If two polygons are similar, then the ratios of the lengths of the two corresponding sides is called the scale factor.

Example 2 Determine Whether Polygons are Similar

Determine whether the triangles are similar. If they are similar, write a similarity statement and find the scale factor of Figure B to Figure A.

SOLUTION

Check whether the corresponding angles are congruent.

1.

From the diagram, you can see that G M, H K, and J L. Therefore, the corresponding angles are congruent.

Example 2 Determine Whether Polygons are Similar

GHMK

912

9 ÷ 312 ÷ 3= = 3

4=

HJKL

1216

12 ÷ 416 ÷ 4= = 3

4=

JGLM

1520

15 ÷ 520 ÷ 5= = 3

4=

All three ratios are equal, so the corresponding side lengths are proportional.

ANSWER By definition, the triangles are similar.GHJ ~ MKL. The scale factor of Figure B to Figure A is 3

4 .

Check whether the corresponding side lengths are proportional.2.

Now You Try Determine Whether Polygons are Similar

Determine whether the polygons are similar. If they are similar, write a similarity statement and find the scale factor of Figure B to Figure A.

1.

2.

ANSWER yes; XYZ ~ DEF;23

ANSWER no69

1012

≠

Example 3 Use Similar Polygons

RST ~ GHJ.

Find the value of x.

SOLUTION

Because the triangles are similar, the corresponding side lengths are proportional. To find the value of x, you can use the following proportion.

RSGH

TRJG

= Write proportion.

1015

x9

=Substitute 15 for GH, 10 for RS,9 for JG, and x for TR.

Cross product property15 · x = 10 · 9

Example 3 Use Similar Polygons

1515x

1590= Divide each side by 15.

15x = 90 Multiply.

Simplify.x = 6

Example 4 Perimeters of Similar Polygons

The outlines of a pool and the patio around the pool are similar rectangles.

Find the ratio of the lengthof the patio to the length ofthe pool.

a.

Find the ratio of theperimeter of the patio tothe perimeter of the pool.

b.

The ratio of the length of the patio to the length of the pool isa.

SOLUTION

length of poollength of the patio

32 feet48 feet= 32 ÷ 16

48 ÷ 16= = .23

Example 4 Perimeters of Similar Polygons

perimeter of poolperimeter of patio

96 feet144 feet= 96 ÷ 48

144 ÷ 48= = .23

b. The perimeter of the patio is 2(24) + 2(48) = 144 feet.

The perimeter of the pool is 2(16) + 2(32) = 96 feet.

The ratio of the perimeter of the patio to the perimeter of the pool is

Now You Try Use Similar Polygons

ANSWER 18

ANSWER21Find the ratio of the perimeter of

STU to the perimeter of PQR.4.

3. Find the value of x.

In the diagram, PQR ~ STU.

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