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9.18: Unknown Measures in Similar Figures

Difficulty Level: At Grade Created by: CK-12
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Practice Unknown Measures of Similar Figures

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Jurnee wants to make a new sail for her father's sailboat. She finds the old sketch that she used to make the first sail and she redraws the sketch so that she can make the new sail. She uses inches as her scale for the figures, but she plans to use feet when she makes the actual sail. Her two figures are similar.

How can Jurnee use a proportion to decide the length of \begin{align*}\overline{KJ}\end{align*} in inches?

In this concept, you will learn how to use proportions to figure out the length of a missing side.

Calculating Unknown Measures in Similar Figures

You can write ratios to compare the lengths of sides.

First, identify the corresponding sides of these two similar triangles, then place the first side in the numerator and the corresponding side in the denominator.

\begin{align*}\frac{LM}{OP} = \frac{LN}{OQ} = \frac{MN}{PQ}\end{align*}

These ratios are written in a proportion or a set of three equal ratios. Remember that there is a relationship between the corresponding sides because they are parts of similar triangles. The side lengths of the similar triangles form a proportion.

Let’s substitute the given measurements into the formula.

\begin{align*}\frac{6}{3} = \frac{8}{4} = \frac{4}{2}\end{align*}

There is a pattern with the ratios of corresponding sides. You can see that the measurement of each side of the first triangle divided by two is the measure of the corresponding side of the second triangle.

Use patterns like this to problem solve the length of missing sides of similar triangles.

These are two similar triangles because they have the same shape but a different size. Therefore, the corresponding sides are similar.

If you look at the side lengths, you should see that there is one variable. That is the missing side length. You can figure out the missing side length by using proportions because the corresponding side lengths form a proportion. Let’s write ratios that form a proportion and find the pattern to figure out the length of the missing side.

\begin{align*}\frac{AB}{DE} & = \frac{AC}{DF} = \frac{BC}{EF} \\ \frac{5}{10} & = \frac{15}{x} = \frac{10}{20}\end{align*}

Looking at this you can see the pattern. The side lengths of the second triangle are double the length of the corresponding side of the first triangle.

Using this pattern, you can see that the length of \begin{align*}DF\end{align*} in the second triangle will be twice the length of \begin{align*}AC\end{align*}. The length of \begin{align*}AC\end{align*} is 15.

15 \begin{align*}\times\end{align*} 2 \begin{align*}=\end{align*} 30

The length of \begin{align*}DF\end{align*} is 30.

Examples

Example 1

Earlier, you were given a problem about Jurnee and the sail.

She wants to make a new sail for her father. She uses an old sketch and makes a new sketch that is similar.

How can Jurnee decide the length of  \begin{align*}\overline{KJ}\end{align*} in inches?

First, use the corresponding sides to set up a proportion.

\begin{align*}\frac{\overline{KJ}}{5} = \frac{6}{4}\end{align*}

Next, use cross products.

\begin{align*}\overline{KJ} \times 4 &= 4\overline{KJ}\\ 5 \times 6 &= 30\\ 4\overline{KJ} &= 30\end{align*}

Then, solve for  \begin{align*}\overline{KJ}\end{align*}.

\begin{align*}30 \div 4 &= 7.5\\ \overline{KJ}&= 7.5\end{align*}

The answer is that \begin{align*}\overline{KJ}=7.5\end{align*} cm.

Example 2

Solve for the missing value.

\begin{align*}\frac{8}{10} = \frac{4}{5} = \frac{2}{x}\end{align*}

First, identify the pattern.

The denominator is the numerator divided by 0.8.

Next, set up an equation to solve for \begin{align*}x\end{align*}.

\begin{align*}\frac{2}{0.8}=x\end{align*}

Then, solve for \begin{align*}x\end{align*}.

\begin{align*}x=2.5\end{align*}

The answer is that \begin{align*}x = 2.5\end{align*}.

Example 3

Solve for the missing value.

\begin{align*}\frac{6}{12} = \frac{x}{24} = \frac{3}{6}\end{align*}

First, identify the pattern.

The denominator is twice the size of the numerator.

Next, set up an equation to solve for \begin{align*}x\end{align*}.

\begin{align*}24 = 2 \times x \end{align*}

Then, solve for \begin{align*}x\end{align*}.

\begin{align*}\frac{24}{2}=x=12\end{align*}

The answer is that \begin{align*}x = 12\end{align*}.

Example 4

Solve for the missing value.

\begin{align*}\frac{12}{x} = \frac{16}{4} = \frac{20}{5}\end{align*}

First, identify the pattern.

The denominator is the result of dividing the numerator by 4.

Next, set up an equation to solve for \begin{align*}x\end{align*}.

\begin{align*}\frac{12}{4}=x\end{align*}

Then, solve for \begin{align*}x\end{align*}.

\begin{align*}x=3\end{align*}

The answer is that \begin{align*}x = 3\end{align*}.

Example 5

Solve for the missing value.

\begin{align*}\frac{8}{2} = \frac{16}{4} = \frac{x}{1}\end{align*}

First, identify the pattern.

The denominator is the result of dividing the numerator by 4.

Next, set up an equation to solve for \begin{align*}x\end{align*}.

\begin{align*}\frac{x}{4}=1\end{align*}

Then, solve for \begin{align*}x\end{align*}.

\begin{align*}x=4\times 1=4\end{align*}

The answer is that \begin{align*}x = 4\end{align*}.

Review

Use the figures to answer the following questions.

1. Are these two triangles similar or congruent?
2. How do you know?
3. Which side is congruent to \begin{align*}AB\end{align*}?
4. Which side is congruent to \begin{align*}AC\end{align*}?
5. Which side is congruent to \begin{align*}RS\end{align*}?
6. Look at the following proportion and solve for missing side length \begin{align*}x\end{align*}.

\begin{align*}\frac{7}{3.5} & = \frac{x}{3.5} = \frac{6}{y} \\ x & = \underline{\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

1. What is the side length for \begin{align*}y\end{align*}?
2. How did you figure these out?

Figure out the missing value in each pair of ratios.

1. \begin{align*}\frac{6}{12} = \frac{x}{24}\end{align*}
2. \begin{align*}\frac{8}{12} = \frac{x}{3}\end{align*}
3. \begin{align*}\frac{9}{10} = \frac{18}{y}\end{align*}
4. \begin{align*}\frac{4}{5} = \frac{x}{2.5}\end{align*}
5. \begin{align*}\frac{16}{20} = \frac{4}{y}\end{align*}
6. \begin{align*}\frac{19}{21} = \frac{x}{42}\end{align*}
7. \begin{align*}\frac{9}{54} = \frac{6}{y}\end{align*}

To see the Review answers, open this PDF file and look for section 9.18.

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