7.2: Similar Polygons
Learning Objectives
 Recognize similar polygons.
 Identify corresponding angles and sides of similar polygons from a similarity statement.
 Use scale factors.
Review Queue
 Solve the proportions.

6x=1015 
47=2x+142 
58=x−22x

 In the picture,
ABXZ=BCXY=ACYZ . Find
AB .  Find
BC .  What is
AB:XZ ?
 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?
Similar Polygons
Think about similar polygons as enlarging or shrinking the same shape. The symbol
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
Solution: Just like a congruence statement, the congruent angles line up within the statement. So,
Because of the corollaries we learned in the last section, the proportions in Example 1 could be written several different ways. For example,
Example 2:
Solution: In the similarity statement,
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:
Solution: All the corresponding angles are congruent because the shapes are rectangles.
Let’s see if the sides are proportional.
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
Solution: 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.
Example 5:
Solution: Line up the corresponding sides.
Example 6: Find the perimeters of
Solution: Perimeter of
Perimeter of
The ratio of the perimeters is 140:210, which reduces to 2:3.
Theorem 72: 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:
Solution: From the similarity statement,
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 18 use the definition of similarity and different types of polygons.
 Questions 913 are similar to Examples 1, 5, 6, and 7.
 Questions 14 and 15 are similar to the Know What?
 Questions 1620 are similar to Example 2.
 Questions 2130 are similar to Examples 3 and 4.
For questions 18, 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.
 \begin{align*}\triangle BIG \sim \triangle HAT\end{align*}. List the congruent angles and proportions for the sides.
 If \begin{align*}BI = 9\end{align*} and \begin{align*}HA = 15\end{align*}, find the scale factor.
 If \begin{align*}BG = 21\end{align*}, find \begin{align*}HT\end{align*}.
 If \begin{align*}AT = 45\end{align*}, find \begin{align*}IG\end{align*}.
 Find the perimeter of \begin{align*}\triangle BIG\end{align*} and \begin{align*}\triangle HAT\end{align*}. 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. NonHD TVs have sides in a ratio of 4:3. Are these two ratios equivalent?
Use the picture to the right to answer questions 1620.
 Find \begin{align*}m \angle E\end{align*} and \begin{align*}m \angle Q\end{align*}.
 \begin{align*}ABCDE \sim QLMNP\end{align*}, find the scale factor.
 Find \begin{align*}BC\end{align*}.
 Find \begin{align*}CD\end{align*}.
 Find \begin{align*}NP\end{align*}.
Determine if the following triangles and quadrilaterals are similar. If they are, write the similarity statement.
Review Queue Answers
 \begin{align*}x = 9\end{align*}
 \begin{align*}x = 11.5\end{align*}
 \begin{align*}x = 8\end{align*}
 \begin{align*}AB = 16\end{align*}
 \begin{align*}BC = 14\end{align*}
 \begin{align*}\frac{2}{3}\end{align*}