What if you were told that two pentagons were similar and you were given the lengths of each pentagon's sides. How could you determine the scale factor of pentagon #1 to pentagon #2? After completing this Concept, you'll be able to answer questions like this one about similar polygons.

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CK-12 Similar Polygons and Scale Factors

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### Guidance

**
Similar polygons
**
are two polygons with the same shape, but not the same size. Similar polygons have corresponding angles that are
**
congruent,
**
and corresponding sides that are
**
proportional.
**

These polygons are not similar:

Think about similar polygons as enlarging or shrinking the same shape. The symbol
is used to represent similarity. Specific types of triangles, quadrilaterals, and polygons will always be similar. For example,
**
all equilateral triangles are similar
**
and

**If two polygons are similar, we know the lengths of corresponding sides are proportional. In similar polygons, the ratio of one side of a polygon to the corresponding side of the other is called the**

*all squares are similar.***scale factor**. The ratio of all parts of a polygon (including the perimeters, diagonals, medians, midsegments, altitudes) is the same as the ratio of the sides.

#### Example A

Suppose . Based on the similarity statement, which angles are congruent and which sides are proportional?

Just like in a congruence statement, the congruent angles line up within the similarity statement. So, and . Write the sides in a proportion: . Note that the proportion could be written in different ways. For example, is also true.

#### Example B

. What are the values of and ?

In the similarity statement, , so . For and , set up proportions.

#### Example C

. Find the scale factor and the length of .

Line up the corresponding sides, and , so the scale factor is or . Because is in the bigger rectangle, we will multiply 40 by because is greater than 1. .

CK-12 Similar Polygons and Scale Factors

### Guided Practice

1. and are below. Are these two rectangles similar?

2. What is the scale factor of to ? Write the similarity statement.

3. . The perimeter of is 150, and . Find the perimeter of .

**
Answers:
**

1. All the corresponding angles are congruent because the shapes are rectangles.

Let’s see if the sides are proportional.
and
.
, so the sides are
**
not
**
in the same proportion, and the rectangles are

**similar.**

*not*2. All the sides are in the same ratio. Pick the two largest (or smallest) sides to find the ratio.

For the similarity statement, line up the proportional sides. so .

3. From the similarity statement, and are corresponding sides. The scale factor is or . is the smaller triangle, so the perimeter of is .

### Explore More

For questions 1-8, determine whether the following statements are true or false.

- All equilateral triangles are similar.
- All isosceles triangles are similar.
- All rectangles are similar.
- All rhombuses are similar.
- All squares are similar.
- All congruent polygons are similar.
- All similar polygons are congruent.
- All regular pentagons are similar.
- . List the congruent angles and proportions for the sides.
- If and , find the scale factor.
- If , find .
- If , find .
- Find the perimeter of and . What is the ratio of the perimeters?
- An NBA basketball court is a rectangle that is 94 feet by 50 feet. A high school basketball court is a rectangle that is 84 feet by 50 feet. Are the two rectangles similar?
- HD TVs have sides in a ratio of 16:9. Non-HD TVs have sides in a ratio of 4:3. Are these two ratios equivalent?

Use the picture to the right to answer questions 16-20.

- Find and .
- , find the scale factor.
- Find .
- Find .
- Find .

Determine if the following triangles and quadrilaterals are similar. If they are, write the similarity statement.