# SIMILAR POLYGONS

## 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.

In the diagram shown below, PQRS is similar to WXYZ. The symbol  is used to indicate similarity.

So, we have

PQRS ∼ WXYZ

In the diagram shown above, because PQRS is similar to WXYZ, we have

PQ / WX  =  QR / XY  =  RS / YZ  =  SP / ZW

## Similar Polygons - Theorem

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

In the diagram shown above, if KLMN ∼ PQRS, then

## Writing Similarity Statements

Example :

Pentagons JKLMN and STUVW shown below are similar. List all the pairs of congruent angles. Write the ratios of the corresponding sides in a statement of proportionality.

Solution :

Because JKLM ∼ STUVW, we can write

∠J  ≅  ∠S

∠K  ≅  ∠T

∠L  ≅  ∠U

∠M  ≅  ∠V

∠N  ≅  ∠W

We can write the statement of proportionality as follows :

JK / ST  =  KL / TU  =  LM / UV  =  MN / VW  =  NJ / WS

## Comparing Similar Polygons

Example :

Decide whether the figures are similar. If they are similar, write a similarity statement.

As shown, the corresponding angles of ABCD and EFGH are congruent. Also, the corresponding side lengths are proportional.

AB / EF  =  15 / 10  =  3 / 2

BC / FG  =  6 / 4  =  3 /2

CD / GH  =  9 /6  =  3 / 2

DA / HE  =  12 / 8  =  3 / 2

Hence, the two figures are similar and we can write

ABCD ∼ EFGH

Note :

If two polygons are similar, then the ratio of the lengths of two corresponding sides is called the scale factor. In the above example, the common ratio is 3/2 is the scale factor of ABCD to EFGH.

## Comparing Photographic Enlargements

Example :

We are asked to create a poster to advertise a field trip to see the Liberty Bell. We have a 3.5 inch by 5 inch photo that you want to enlarge. We want the enlargement to be 16 inches wide. How long will it be ?

Solution :

To find the length of the enlargement, you can compare the enlargement to the original measurements of the photo.

From the diagram shown above, we have

x in. / 5 in.  =  16 in. / 3.5 in.

5 ⋅ (x in. / 5 in.  =  16 in. / 3.5 in.) ⋅ 5

Simplify.

x  ≈  22.9 in.

Hence, the length of the enlargement will be about 23 inches.

## Using Similar Polygons

Example 1 :

The rectangular patio around a pool is similar to the pool as shown below. Calculate the scale factor of the patio to the pool, and find the ratio of their perimeters.

Solution :

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

=  144 /  96

=  3 / 2

In similar figures, the ratio of the perimeters is the same as the scale factor.

So, the scale factor of the patio to the pool is 3/2.

Example 2 :

In the diagram shown below, quadrilateral JKLM is similar to quadrilateral PQRS. Find the value of z.

Solution :

Because the quadrilaterals JKLM is similar to PQRS, we can set up the proportion that contains PQ.

Write proportion :

KL / QR  =  JK / PQ

Substitute.

15 / 6  =  10 / z

5 / 2  =  10 / z

Using reciprocal property,

2 / 5  =  z / 10

or

z / 10  =  2 / 5

Multiply each side by 10.

10 ⋅ (z / 10)  =  10 ⋅ (2 / 5)

Simplify.

z  =  4

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