# 7.2: Similar Polygons

## Learning Objectives

- Recognize similar polygons.
- Identify corresponding angles and sides of similar polygons from a similarity statement.
- Calculate and apply scale factors.

## Review Queue

- Solve the proportions.
- In the picture, .
- Find .
- Find .

**Know What?** A baseball diamond is a square with 90 foot sides. A softball diamond is a square with 60 foot sides. Are the two diamonds similar? If so, what is the scale factor? Explain your answer.

## Similar Polygons

**Similar Polygons:** Two polygons with the same shape, but not the same size.

Think about similar polygons as an enlargement or shrinking of the same shape. So, more specifically, similar polygons have to have the same number of sides, the corresponding angles are congruent, and the corresponding sides are proportional. The symbol is used to represent similar. Here are some examples:

These polygons are not similar:

**Example 1:** Suppose . Based on the similarity statement, which angles are congruent and which sides are proportional?

**Solution:** Just like a congruence statement, the congruent angles line up within the statement. So, and . The same is true of the proportional sides. We write the sides in a proportion, .

Because of the corollaries we learned in the last section, the proportions in Example 1 could be written several different ways. For example, . Make sure to line up the corresponding proportional sides.

**Example 2:** . What are the values of and ?

**Solution:** In the similarity statement, , so . For and , set up a proportion.

Specific types of triangles, quadrilaterals, and polygons will always be similar. For example, because all the angles and sides are congruent, ** all equilateral triangles are similar.** For the same reason,

**We can take this one step further and say that all regular polygons (with the same number of sides) are similar.**

*all squares are similar.*
**Example 3:** is a rectangle with length 12 and width 8. is a rectangle with length 24 and width 18. Are these two rectangles similar?

**Solution:** Draw a picture. First, all the corresponding angles need to be congruent. In rectangles, all the angles are congruent, so this condition is satisfied. Now, let’s see if the sides are proportional. . . This tells us that the sides are not in the same proportion, so the rectangles are not similar. We could have also set up the proportion as and . , so you would end up with the same conclusion.

## Scale Factors

If two polygons are similar, we know the lengths of corresponding sides are proportional. If is the length of a side in one polygon, and is the length of the corresponding side in the other polygon, then the ratio is the ** scale factor** relating the first polygon to the second.

**Scale Factor:** In similar polygons, the ratio of one side of a polygon to the corresponding side of the other.

**Example 5:** . Find the scale factor and the length of .

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

**Example 6:** Find the perimeters of and . Then find the ratio of the perimeters.

**Solution:** Perimeter of

Perimeter of

The ratio of the perimeters is 140:210, which reduces to 2:3.

**Theorem 7-2:** The ratio of the perimeters of two similar polygons is the same as the ratio of the sides.

In addition the perimeter being in the same ratio as the sides, all parts of a polygon are in the same ratio as the sides. This includes diagonals, medians, midsegments, altitudes, and others.

**Example 7:** . The perimeter of is 150 and and . Find the perimeter of .

**Solution:** From the similarity statement, and are corresponding sides. So, the scale factor is or . The perimeter of is .

**Know What? Revisited** All of the sides in the baseball diamond are 90 feet long and 60 feet long in the softball diamond. This means all the sides are in a ratio. All the angles in a square are congruent, all the angles in both diamonds are congruent. The two squares are similar and the scale factor is .

## Review Questions

Determine if 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?

Use the picture to the right to answer questions 14-18.

- 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.

- Solve for and .
- Find the perimeter of .
- Solve for and .
- Solve for .
- Solve for and .
- Trapezoids Solve for and .
- Two similar octagons have a scale factor of . If the perimeter of the smaller octagon is 99 meters, what is the perimeter of the larger octagon?
- Two right triangles are similar. The legs of one of the triangles are 5 and 12. The second right triangle has a hypotenuse of length 39. What is the scale factor between the two triangles?
- What is the area of the smaller triangle in problem 30? What is the area of the larger triangle in problem 30? What is the ratio of the areas? How does it compare to the ratio of the lengths (or scale factor)? Recall that the area of a triangle is .