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.

6x=1015 
47=2x+142 
58=x−22x

 In the picture,
ABXZ=BCXY=ACYZ . Find
AB .  Find
BC .
 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
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, because all the angles and sides are congruent, all equilateral triangles are similar. For the same reason, all squares are similar. We can take this one step further and say that all regular polygons (with the same number of sides) are similar.
Example 3:
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.
Scale Factors
If two polygons are similar, we know the lengths of corresponding sides are proportional. If
Scale Factor: In similar polygons, the ratio of one side of a polygon to the corresponding side of the other.
Example 5:
Solution: Line up the corresponding proportional 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 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:
Solution: From the similarity statement,
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 \begin{align*}\frac{90}{60}=\frac{3}{2}\end{align*} 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 \begin{align*}\frac{3}{2}\end{align*}.
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.
 \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?
Use the picture to the right to answer questions 1418.
 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.
 \begin{align*}\triangle ABC \sim \triangle DEF\end{align*} \begin{align*}{\;}\end{align*} Solve for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.
 \begin{align*}QUAD \sim KENT\end{align*} \begin{align*}{\;}\end{align*} Find the perimeter of \begin{align*}QUAD\end{align*}.
 \begin{align*}\triangle CAT \sim \triangle DOG\end{align*} \begin{align*}{\;}\end{align*} Solve for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.
 \begin{align*}PENTA \sim FIVER\end{align*} \begin{align*}{\;}\end{align*} Solve for \begin{align*}x\end{align*}.
 \begin{align*}\triangle MNO \sim \triangle XNY\end{align*} \begin{align*}{\;}\end{align*} Solve for \begin{align*}a\end{align*} and \begin{align*}b\end{align*}.
 Trapezoids \begin{align*}HAVE \sim KNOT\end{align*} Solve for \begin{align*}x\end{align*} and \begin{align*}y\end{align*}.
 Two similar octagons have a scale factor of \begin{align*}\frac{9}{11}\end{align*}. 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 \begin{align*}A=\frac{1}{2} \ bh\end{align*}.
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*}
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