# 7.2: Similar Polygons

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Recognize similar polygons.
• Identify corresponding angles and sides of similar polygons from a similarity statement.
• Use scale factors.

## Review Queue

1. Solve the proportions.
1. \begin{align*}\frac{6}{x} = \frac{10}{15}\end{align*}
2. \begin{align*}\frac{4}{7} = \frac{2x+1}{42}\end{align*}
3. \begin{align*}\frac{5}{8} = \frac{x-2}{2x}\end{align*}
2. In the picture, \begin{align*}\frac{AB}{XZ} = \frac{BC}{XY} = \frac{AC}{YZ}\end{align*}.
1. Find \begin{align*}AB\end{align*}.
2. Find \begin{align*}BC\end{align*}.
3. What is \begin{align*}AB:XZ\end{align*}?

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?

## Similar Polygons

Think about similar polygons as enlarging or shrinking the same shape. The symbol \begin{align*}\sim\end{align*} is used to represent similar.

Similar Polygons: Two polygons with the same shape, but not the same size. The corresponding angles are congruent, and the corresponding sides are proportional.

These polygons are not similar:

Example 1: Suppose \begin{align*}\triangle ABC \sim \triangle JKL\end{align*}. 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, \begin{align*}\angle A \cong \angle J, \angle B \cong \angle K,\end{align*} and \begin{align*}\angle C \cong \angle L\end{align*}. Write the sides in a proportion, \begin{align*}\frac{AB}{JK} = \frac{BC}{KL} = \frac{AC}{JL}\end{align*}.

Because of the corollaries we learned in the last section, the proportions in Example 1 could be written several different ways. For example, \begin{align*}\frac{AB}{BC} = \frac{JK}{KL}\end{align*} is also true.

Example 2: \begin{align*}MNPQ \sim RSTU\end{align*}. What are the values of \begin{align*}x, y\end{align*} and \begin{align*}z\end{align*}?

Solution: In the similarity statement, \begin{align*}\angle M \cong \angle R\end{align*}, so \begin{align*}z = 115^{\circ}\end{align*}. For \begin{align*}x\end{align*} and \begin{align*}y\end{align*}, set up a proportion.

\begin{align*}\frac{18}{30} &= \frac{x}{25} && \ \frac{18}{30} = \frac{15}{y}\\ 450 &= 30x && 450 = 18y\\ x &= 15 && \quad y = 25\end{align*}

Specific types of triangles, quadrilaterals, and polygons will always be similar. For example, all equilateral triangles are similar and all squares are similar.

Example 3: \begin{align*}ABCD\end{align*} and \begin{align*}UVWX\end{align*} are below. Are these two rectangles similar?

Solution: All the corresponding angles are congruent because the shapes are rectangles.

Let’s see if the sides are proportional. \begin{align*}\frac{8}{12} = \frac{2}{3}\end{align*} and \begin{align*}\frac{18}{24} = \frac{3}{4}\end{align*}. \begin{align*}\frac{2}{3} \neq \frac{3}{4}\end{align*}, so the sides are not in the same proportion, so the rectangles are not similar.

## Scale Factors

If two polygons are similar, we know the lengths of corresponding sides are proportional.

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

Example 4: What is the scale factor of \begin{align*}\triangle ABC\end{align*} to \begin{align*}\triangle XYZ\end{align*}? Write the similarity statement.

Solution: All the sides are in the same ratio. Pick the two largest (or smallest) sides to find the ratio.

\begin{align*}\frac{15}{20} = \frac{3}{4}\end{align*}

For the similarity statement, line up the proportional sides. \begin{align*}AB \rightarrow XY, BC \rightarrow XZ, AC \rightarrow YZ,\end{align*} so \begin{align*}\triangle ABC \sim \triangle YXZ\end{align*}.

Example 5: \begin{align*}ABCD \sim AMNP\end{align*}. Find the scale factor and the length of \begin{align*}BC\end{align*}.

Solution: Line up the corresponding sides. \begin{align*}AB:AM\end{align*}, so the scale factor is \begin{align*}\frac{30}{45} = \frac{2}{3}\end{align*} or \begin{align*}\frac{3}{2}\end{align*}. Because \begin{align*}BC\end{align*} is in the bigger rectangle, we will multiply 40 by \begin{align*}\frac{3}{2}\end{align*} because it is greater than 1. \begin{align*}BC = \frac{3}{2} (40)=60\end{align*}.

Example 6: Find the perimeters of \begin{align*}ABCD\end{align*} and \begin{align*}AMNP\end{align*}. Then find the ratio of the perimeters.

Solution: Perimeter of \begin{align*}ABCD = 60 + 45 + 60 + 45 = 210\end{align*}

Perimeter of \begin{align*}AMNP = 40 + 30 + 40 + 30 = 140\end{align*}

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 to the perimeter having 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: \begin{align*}\triangle ABC \sim \triangle MNP\end{align*}. The perimeter of \begin{align*}\triangle ABC\end{align*} is 150, \begin{align*}AB = 32\end{align*} and \begin{align*}MN = 48\end{align*}. Find the perimeter of \begin{align*}\triangle MNP\end{align*}.

Solution: From the similarity statement, \begin{align*}AB\end{align*} and \begin{align*}MN\end{align*} are corresponding sides. The scale factor is \begin{align*}\frac{32}{48} = \frac{2}{3}\end{align*}. \begin{align*}\triangle ABC\end{align*} is the smaller triangle, so the perimeter of \begin{align*}\triangle MNP\end{align*} is \begin{align*}\frac{3}{2} (150)=225\end{align*}.

Know What? Revisited The baseball diamond is on the left and the softball diamond is on the right. All the angles and sides are congruent, so all squares are similar. All of the sides in the baseball diamond are 90 feet long and 60 feet long in the softball diamond. This means the scale factor is \begin{align*}\frac{90}{60} = \frac{3}{2}\end{align*}.

## Review Questions

• Questions 1-8 use the definition of similarity and different types of polygons.
• Questions 9-13 are similar to Examples 1, 5, 6, and 7.
• Questions 14 and 15 are similar to the Know What?
• Questions 16-20 are similar to Example 2.
• Questions 21-30 are similar to Examples 3 and 4.

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

1. All equilateral triangles are similar.
2. All isosceles triangles are similar.
3. All rectangles are similar.
4. All rhombuses are similar.
5. All squares are similar.
6. All congruent polygons are similar.
7. All similar polygons are congruent.
8. All regular pentagons are similar.
9. \begin{align*}\triangle BIG \sim \triangle HAT\end{align*}. List the congruent angles and proportions for the sides.
10. If \begin{align*}BI = 9\end{align*} and \begin{align*}HA = 15\end{align*}, find the scale factor.
11. If \begin{align*}BG = 21\end{align*}, find \begin{align*}HT\end{align*}.
12. If \begin{align*}AT = 45\end{align*}, find \begin{align*}IG\end{align*}.
13. Find the perimeter of \begin{align*}\triangle BIG\end{align*} and \begin{align*}\triangle HAT\end{align*}. What is the ratio of the perimeters?
14. 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?
15. 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.

1. Find \begin{align*}m \angle E\end{align*} and \begin{align*}m \angle Q\end{align*}.
2. \begin{align*}ABCDE \sim QLMNP\end{align*}, find the scale factor.
3. Find \begin{align*}BC\end{align*}.
4. Find \begin{align*}CD\end{align*}.
5. Find \begin{align*}NP\end{align*}.

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

1. \begin{align*}x = 9\end{align*}
2. \begin{align*}x = 11.5\end{align*}
3. \begin{align*}x = 8\end{align*}
1. \begin{align*}AB = 16\end{align*}
2. \begin{align*}BC = 14\end{align*}
3. \begin{align*}\frac{2}{3}\end{align*}

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