What if you were told that two pentagons were similar and you were given the lengths of each pentagon's sides. How could you determine the scale factor of pentagon #1 to pentagon #2? After completing this Concept, you'll be able to answer questions like this one about similar polygons.

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CK-12 Similar Polygons and Scale Factors

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### Guidance

**Similar polygons** are two polygons with the same shape, but not 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 \begin{align*}\sim\end{align*} is used to represent similarity. Specific types of triangles, quadrilaterals, and polygons will always be similar. For example, ** all equilateral triangles are similar** and

**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**

*all squares are similar.***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 \begin{align*}\triangle ABC \sim \triangle JKL\end{align*}. 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, \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*}. Note that the proportion could be written in different ways. For example, \begin{align*}\frac{AB}{BC} = \frac{JK}{KL}\end{align*} is also true.

#### Example B

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

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 proportions.

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

#### Example C

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

Line up the corresponding sides, \begin{align*}AB\end{align*} and \begin{align*}AM = CD\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 \begin{align*}\frac{3}{2}\end{align*} is greater than 1. \begin{align*}BC = \frac{3}{2} (40)=60\end{align*}.

CK-12 Similar Polygons and Scale Factors

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### Guided Practice

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

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

3. \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*}.

**Answers:**

1. 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, and the rectangles are

**similar.**

*not*2. 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*}.

3. 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*} or \begin{align*} \frac{3}{2}\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*}.

### Explore More

For questions 1-8, determine whether 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. 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.

- 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.

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 7.3.