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# 8.14: Scale Factor of Similar Polygons

Difficulty Level: At Grade Created by: CK-12
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Carla enjoys making replicas of houses, planes and cars. She is careful to maintain the same proportional relationship between her replica and the original. She finds that knowing the scale factor makes the process much easier. She is currently working on a replica of a house. The scale factor of the house to her replica is 10.5. If the house is 21 feet tall, how tall is her replica?

In this concept, you will learn about the relationship between scale factors of similar polygons.

### Relationship Between Scale Factors of Similar Polygons

Similar figures are shapes that exist in proportion to each other. They have congruent angles, but their sides are different lengths. Squares, for example, are similar to each other because they always have four 90\begin{align*}90^{\circ}\end{align*} angles and four equal sides, even if the lengths of their sides differ. Other figures can be similar too, if their angles are equal. Let’s look at some pairs of similar figures.

Notice that in each pair the figures look the same, but one is smaller than the other. As you can see, similar figures have congruent angles but sides of different lengths.

Each pair of corresponding sides has the same relationship as every other pair of corresponding sides, so that, altogether, the pairs of sides exist in proportion to each other. For instance, if a side in one figure is twice as long as its corresponding side in a similar figure, all of the other sides will be twice as long too.

These relationships can be used to find the measures of unknown sides. This method is called indirect measurement.

Let’s look at similar figures to understand how indirect measurement works.

Similar figures have exactly the same angles. Therefore each angle in one figure corresponds to an angle in the other.

These triangles are similar because their angles have the same measures. Angle B\begin{align*}B\end{align*} is 100\begin{align*}100^{\circ}\end{align*}. Its corresponding angle will also measure 100\begin{align*}100^{\circ}\end{align*}, that makes angle Q\begin{align*}Q\end{align*} its corresponding angle. Angles A\begin{align*}A\end{align*} and P\begin{align*}P\end{align*} correspond, and angles C\begin{align*}C\end{align*} and R\begin{align*}R\end{align*} correspond.

Similar figures also have corresponding sides, even though the sides are not congruent. Corresponding sides are not always easy to spot. You can think of corresponding sides as those which are in the same place in relation to corresponding angles. For instance, side AB\begin{align*}AB\end{align*}, between angles A\begin{align*}A\end{align*} and B\begin{align*}B\end{align*}, must correspond to side PQ\begin{align*}PQ\end{align*}, because A\begin{align*}A\end{align*} corresponds to P\begin{align*}P\end{align*} and B\begin{align*}B\end{align*} corresponds to Q\begin{align*}Q\end{align*}.

Corresponding sides also have lengths that are related, even though they are not congruent. Specifically, the side lengths are proportional. In other words, each pair of corresponding sides has the same ratio as every other pair of corresponding sides. Look at the example below.

These rectangles are similar because the sides of one are proportional to the other. You can see this if you set up proportions for each pair of corresponding sides. Let’s put the sides of the large rectangle on the top and the corresponding sides of the small rectangle on the bottom. It doesn’t matter which is put on top, as long as you keep all the sides from one figure in the same place.

LMWXMNXYONZYLOWZ=84=63=84=63\begin{align*}\frac{LM}{WX} &= \frac{8}{4}\\ \frac{MN}{XY} &= \frac{6}{3}\\ \frac{ON}{ZY} &= \frac{8}{4}\\ \frac{LO}{WZ} &= \frac{6}{3}\end{align*}

Now you can clearly see each relationship. To figure out if the pairs do indeed form a proportion, you have to divide the numerator by the denominator. If the quotient is the same, then the ratios each form the same proportion and the figures are similar.

LMWXMNXYONZYLOWZ=84=2=63=2=84=2=63=2\begin{align*}\frac{LM}{WX} &= \frac{8}{4} = 2\\ \frac{MN}{XY} &= \frac{6}{3} = 2\\ \frac{ON}{ZY} &= \frac{8}{4} = 2\\ \frac{LO}{WZ} &= \frac{6}{3} = 2\end{align*}

Each quotient is the same so these ratios are proportional. These quotients are scale factors

The scale factor is the ratio that determines the proportional relationship between the sides of similar figures. For the pairs of sides to be proportional to each other, they must have the same scale factor. In other words, similar figures have congruent angles and sides with the same scale factor. A scale factor of 2 means that each side of the larger figure is twice as long as its corresponding side is in the smaller figure.

### Examples

#### Example 1

Earlier, you were given a problem about Carla and her replica of the house.

She knows that the scale factor of the house to her replica is 10.5. If the house is 21 feet tall, how tall is her replica?

First, set up an equation that reflects the relationship between the heights and the scale factor.

21x=10.5\begin{align*}\frac{21}{x} = 10.5\end{align*}

Then, solve for the missing height.

x=2\begin{align*}x = 2\end{align*}

The answer is that the replica is 2 feet tall.

#### Example 2

What is the scale factor of the figures below?



First, set up the proportions of the sides. Put all the sides from the large figure on top and the sides from the small figure on the bottom.

QRHITSKJRSIJQTHK=155=217=63=155\begin{align*}\frac{QR}{HI} &= \frac{15}{5}\\ \frac{TS}{KJ} &= \frac{21}{7}\\ \frac{RS}{IJ} &= \frac{6}{3}\\ \frac{QT}{HK} &= \frac{15}{5}\end{align*}

Next, divide to find the scale factor.

QRHITSKJRSIJQTHK=155=3=217=3=63=3=155=3\begin{align*}\frac{QR}{HI} &= \frac{15}{5} = 3\\ \frac{TS}{KJ} &= \frac{21}{7} = 3\\ \frac{RS}{IJ} &= \frac{6}{3} = 3\\ \frac{QT}{HK} &= \frac{15}{5} = 3\end{align*}

Then, compare the scale factor.

The factors are all 3.

The answer is that the scale factor of the figures is 3.

#### Example 3

The two figures are similar.



The side lengths are listed as follows:

MN = 3 inches QT = 6 inches

NO = 2 inches QR = 4 inches

MP = 4 inches TS = 8 inches

OP = 2 inches RS = 4 inches

Given the relationship between these sides, what scale factor compares the first figure to the second figure?

First, note the order given by the scale factor.

The scale factor compares the first figure to the second figure.

Next, write the proportion of MN to QT.

36\begin{align*}\frac{3}{6}\end{align*}

Then, simplify the proportion.

12\begin{align*}\frac{1}{2}\end{align*}

The scale factor that compares the first figure to the second figure is 12\begin{align*}\frac{1}{2}\end{align*}.

#### Example 4

Given two figures, the quotients of the corresponding sides equal 3, 3, 5 and 4. Are the two figures similar?

First, state the given information

The quotients are 3, 3, 5 and 4.

Next, understand the relationship between the quotients and similarity.

The quotients of corresponding sides must all equal the same number.

Not similar.

The answer is that the two figures are not similar.

#### Example 5

What is the scale factor of the following proportional sides?

186\begin{align*}\frac{18}{6}\end{align*} and 248\begin{align*}\frac{24}{8}\end{align*}

First, find the quotient of the first proportion.

186=3\begin{align*}\frac{18}{6} = 3\end{align*}

Next, find the quotient of the second proportion.

\begin{align*}\frac{24}{8} = 3\end{align*}

Then, compare the two quotients.

Both proportional sides equal 3.

The answer is that the scale factor is 3.

### Review

Find the scale factor of the pairs of similar figures below.

Use each ratio to determine scale factor.

1. \begin{align*}\frac{3}{1}\end{align*}
2. \begin{align*}\frac{8}{2}\end{align*}
3. \begin{align*}\frac{2}{8}\end{align*}
4. \begin{align*}\frac{10}{5}\end{align*}
5. \begin{align*}\frac{12}{4}\end{align*}
6. \begin{align*}\frac{16}{2}\end{align*}
7. \begin{align*}\frac{15}{3}\end{align*}
8. \begin{align*}\frac{24}{4}\end{align*}
9. \begin{align*}\frac{4}{2}\end{align*}
10. \begin{align*}\frac{6}{2}\end{align*}
11. \begin{align*}\frac{3}{9}\end{align*}

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

Perimeter

Perimeter is the distance around a two-dimensional figure.

Proportion

A proportion is an equation that shows two equivalent ratios.

Scale Factor

A scale factor is a ratio of the scale to the original or actual dimension written in simplest form.

Similar

Two figures are similar if they have the same shape, but not necessarily the same size.

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