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# Similar Polygons and Scale Factors

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What if you were comparing a baseball diamond and a softball diamond? 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? After completing this Concept, you'll be able to use your knowledge of similar polygons to answer these questions.

### Guidance

Similar polygons are two polygons with the same shape, but not necessarily 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 $\sim$ is used to represent similarity. Specific types of triangles, quadrilaterals, and polygons will always be similar. For example, all equilateral triangles are similar and all squares are similar. 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 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 $\triangle ABC \sim \triangle JKL$ . 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, $\angle A \cong \angle J, \angle B \cong \angle K,$ and $\angle C \cong \angle L$ . Write the sides in a proportion: $\frac{AB}{JK} = \frac{BC}{KL} = \frac{AC}{JL}$ . Note that the proportion could be written in different ways. For example, $\frac{AB}{BC} = \frac{JK}{KL}$ is also true.

#### Example B

$MNPQ \sim RSTU$ . What are the values of $x, y$ and $z$ ?

In the similarity statement, $\angle M \cong \angle R$ , so $z = 115^{\circ}$ . For $x$ and $y$ , set up proportions.

$\frac{18}{30} &= \frac{x}{25} && \ \frac{18}{30} = \frac{15}{y}\\450 &= 30x && 18y = 450\\x &= 15 && \quad y = 25$

#### Example C

$ABCD \sim AMNP$ . Find the scale factor and the length of $BC$ .

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

Watch this video for help with the Examples above.

#### Concept Problem 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 $\frac{90}{60}=\frac{3}{2}$ 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 $\frac{3}{2}$ .

### Vocabulary

Similar polygons are two polygons with the same shape, but not necessarily the same size. The corresponding angles of similar polygons are congruent (exactly the same) and the corresponding sides are proportional (in the same ratio). In similar polygons, the ratio of one side of a polygon to the corresponding side of the other is called the scale factor .

### Guided Practice

1. $ABCD$ and $UVWX$ are below. Are these two rectangles similar?

2. What is the scale factor of $\triangle ABC$ to $\triangle XYZ$ ? Write the similarity statement.

3. $\triangle ABC \sim \triangle MNP$ . The perimeter of $\triangle ABC$ is 150, $AB = 32$ and $MN = 48$ . Find the perimeter of $\triangle MNP$ .

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

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

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

$\frac{15}{20} = \frac{3}{4}$

For the similarity statement, line up the proportional sides. $AB \rightarrow XY, BC \rightarrow XZ, AC \rightarrow YZ,$ so $\triangle ABC \sim \triangle YXZ$ .

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

### Practice

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. $\triangle BIG \sim \triangle HAT$ . List the congruent angles and proportions for the sides.
10. If $BI = 9$ and $HA = 15$ , find the scale factor.
11. If $BG = 21$ , find $HT$ .
12. If $AT = 45$ , find $IG$ .
13. Find the perimeter of $\triangle BIG$ and $\triangle HAT$ . What is the ratio of the perimeters?

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

1. Find $m \angle E$ and $m \angle Q$ .
2. $ABCDE \sim QLMNP$ , find the scale factor.
3. Find $BC$ .
4. Find $CD$ .
5. Find $NP$ .

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

1. $\triangle ABC \sim \triangle DEF$ ${\;}$ Solve for $x$ and $y$ .
2. $QUAD \sim KENT$ ${\;}$ Find the perimeter of $QUAD$ .
3. $\triangle CAT \sim \triangle DOG$ ${\;}$ Solve for $x$ and $y$ .
4. $PENTA \sim FIVER$ ${\;}$ Solve for $x$ .
5. $\triangle MNO \sim \triangle XNY$ ${\;}$ Solve for $a$ and $b$ .
6. Trapezoids $HAVE \sim KNOT$ Solve for $x$ and $y$ .
7. Two similar octagons have a scale factor of $\frac{9}{11}$ . If the perimeter of the smaller octagon is 99 meters, what is the perimeter of the larger octagon?
8. 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?
9. 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 $A=\frac{1}{2} \ bh$ .