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Indirect Measurement

Measurement using similar triangles.

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Recognizing Similarity
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When walking home from school, Amira walks past a tower that casts a shadow that is 10 feet long. Standing next to it, she notices that she casts a shadow of 2 feet. Amira knows that she is exactly five feet tall. How can she use this information to determine the height of the tower?

In this concept, you will learn to recognize similarity.

Similarity

Congruent means exactly the same, having the same size and shape.

Sometimes, a figure will have the same shape, but not the same size. It will be either smaller or larger than the original figure. When this happens, you say that the two figures are similar. Similar figures have the same shape, but not the same size.

Think about this for a minute, if a figure has the same shape, but not the same size, then there is still a relationship between the two figures. The relationship is created based on the shape being the same.

Let’s start by thinking about angles. With similar figures, each angle of one figure in a similar pair corresponds and is congruent to an angle in the other.

For instance, the top point of one triangle corresponds to the top point of the other triangle in a similar pair. You call these corresponding parts.

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Notice that the angles in the triangles above match in these two triangles. The shape of the triangles is the same, and you can see that the angles do match.

Now, let’s look at side lengths. The sides in similar pairs also correspond to each other (such as the base of each triangle), but they are not congruent; they are proportional. You can determine whether figures are similar to each other by comparing their corresponding parts. Corresponding parts are especially helpful when one figure is rotated so that it is not clear which angles and sides correspond to which in the other figure.

Look at the image below of the two rectangles.

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Let’s look at the corresponding side lengths.

In the first rectangle, the short side is 4 and the long side is 8. You know that opposite sides of a rectangle are congruent, so you don’t need to worry about writing measurements on the other two sides.

In the second rectangle, the short side is 2 and the long side is 4.

You can compare the measurements in the first rectangle with the ones in the second rectangle.

First, let’s write a proportion to compare the corresponding sides.

\begin{align*}\begin{array}{rcl} \frac{\text{short side}}{\text{short side}} &=& \frac{\text{long side}}{\text{long side}} \\ \frac{4}{2} &=& \frac{8}{4} \end{array}\end{align*}

Next, you can see that these two ratios form a proportion. You can use this information to prove whether or not two figures are similar as well. Remember, the angle measures must be the same, and the side lengths must be proportional.

\begin{align*}\begin{array}{rcl} \frac{4}{2} &=& \frac{8}{4} \\ 2 &=& 2 \end{array}\end{align*}

The answer is that the two rectangles are similar since the angles are congruent (all 90°) and the side lengths are proportional.

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Now that you understand how to identify whether or not two figures are similar, we will look at similar triangles. Similar triangles are very useful because you can use them to figure out measurements.

Many years ago, this is how people used to figure out the measurements for things that were too high or big to measure. They used indirect measurement. Indirect measurement uses similar triangles and proportions to figure out lengths or distances.

First, let’s think about similar triangles.

Similar triangles have the same properties as other similar figures. The angle measures are the same and the corresponding side lengths are proportional.

Let’s look at this diagram to understand this.

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First, let’s compare the angle measures.

\begin{align*}\begin{array}{rcl} \angle A & \cong & \angle D \\ \angle B & \cong & \angle E \\ \angle C & \cong & \angle F \end{array}\end{align*}

Next, you can look at the corresponding side lengths. In the diagram, you haven’t been given any measurements, but you can use the lowercase letters to show which sides correspond.

\begin{align*}\frac{a}{d} = \frac{b}{e} = \frac{c}{f}\end{align*}

This shows that the side lengths form a ratio and that each of these is proportional to the other.

You can use this information when problem solving for missing side lengths. First, we would have to know some of the side lengths.

Let’s assign some lengths to the sides in the diagram above.

\begin{align*}\begin{array}{rcl} a &=& 12 \\ b &=& ? \\ c &=& 3 \\ d &=& 4 \\ e &=& 3 \\ f &=& 1 \end{array} \end{align*}

First, you can take these given measures and substitute them into the proportion that you wrote earlier. Notice that you don’t have the measure of side \begin{align*}b\end{align*}, so you will need to solve for that missing measurement.

\begin{align*}\begin{array}{rcl} && \frac{a}{d} = \frac{b}{e} = \frac{c}{f} \\ && \frac{12}{4} = \frac{b}{3} = \frac{3}{1} \end{array}\end{align*}

Next, you can use two of the three ratios to solve the proportions. You have three ratios, but you don’t need all three because two equal ratios form a proportion. This means that you only need to work with two ratios to solve for the value of \begin{align*}b\end{align*}.

\begin{align*}\begin{array}{rcl} \frac{a}{d} &=& \frac{b}{e} \\ \\ \frac{12}{4} &=& \frac{b}{3} \end{array}\end{align*}

Then, use algebra to solve the proportion.

\begin{align*}\begin{array}{rcl} \frac{12}{4} &=& \frac{b}{3} \\ 4 b &=& 12 \times 3 \\ 4 b &=& 36 \\ \frac{4b}{4} &=& \frac{36}{4} \\ b &=& 9 \end{array}\end{align*}

The answer is 9.

The value of \begin{align*}b\end{align*} is 9.

The key to working with indirect measurement is to always be clear about what is being compared. You write the ratios and then form a proportion to solve for the missing length or distance.

Examples

Example 1

Earlier, you were given a problem about Amira and her shadows.

Notice that the person and the shadow form two sides of a triangle and you can draw an imaginary line from the head of the person to the tip of the shadow. Shadows are a way of working with triangles and indirect measurement. In fact, you will often hear these types of problems referred to as shadow problems.

To solve this one, let’s figure out how to use similar triangles to figure out the height of the tower.

First, think about what is being compared. You are comparing the height of the person, Amira, with the length of the shadow. That is the first ratio.

\begin{align*}\frac{\text{person}}{\text{shadow}} = \frac{5 \ ft}{2 \ ft} \end{align*}

Next, look at the tower. You don’t know the height of the tower that is our variable. You do know the length of the shadow. Here is the second ratio.

\begin{align*}\frac{\text{tower}}{\text{shadow}} = \frac{x \ ft}{10 \ ft}\end{align*}

Then, you can say that these two triangles are similar and that similar triangles are proportional. Therefore, these two ratios form a proportion. Let’s write them as a proportion.

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

Then, use algebra to solve the proportion.

\begin{align*}\begin{array}{rcl} \frac{5}{2} &=& \frac{x}{10} \\ 2x &=& 5 \times 10 \\ 2x &=& 50 \\ \frac{2x}{2} &=& \frac{50}{2} \\ x &=& 25 \end{array}\end{align*}

The answer is 25.

The tower is 25 feet tall.

Example 2

Solve for the missing variable. 

First, you can use two of the three ratios to solve for \begin{align*}x\end{align*}.

\begin{align*}\frac{2}{8} = \frac{24}{x}\end{align*}

Next, use algebra to solve the proportion.

\begin{align*}\begin{array}{rcl} \frac{2}{8} &=& \frac{24}{x} \\ 2 x &=& 24 \times 8 \\ 2x &=& 192 \\ \frac{2x}{2} &=& \frac{192}{2} \\ x &=& 96 \end{array}\end{align*} 

The answer is 96.

The value of \begin{align*}x\end{align*} is 96.

Example 3

Solve for the missing value.

\begin{align*}\frac{2}{3} = \frac{4}{6} = \frac{x}{18}\end{align*}

First, you can use two of the three ratios to solve for \begin{align*}x\end{align*}.

\begin{align*}\frac{4}{6} = \frac{x}{18}\end{align*}

Next, use algebra to solve the proportion.

\begin{align*}\begin{array}{rcl} \frac{4}{6} &=& \frac{x}{18} \\ 6x &=& 4 \times 18 \\ 6x &=& 72 \\ \frac{6x}{6} &=& \frac{72}{6} \\ x &=& 12 \end{array}\end{align*}

The answer is 12.

The value of \begin{align*}x\end{align*} is 12.

Example 4

Solve for the missing value.

\begin{align*}\frac{4}{5} = \frac{12}{15} = \frac{x}{30}\end{align*}

First, you can use two of the three ratios to solve for \begin{align*}x\end{align*}.

\begin{align*}\frac{4}{5} = \frac{x}{30}\end{align*}

Next, use algebra to solve the proportion.

\begin{align*}\begin{array}{rcl} \frac{4}{5} &=& \frac{x}{30} \\ 5x &=& 4 \times 30 \\ 5x &=& 120 \\ \frac{5x}{5} &=& \frac{120}{5} \\ x &=& 24 \end{array}\end{align*}

The answer is 24.

The value of \begin{align*}x\end{align*} is 24.

Example 5

Solve for the missing value.

\begin{align*}\frac{8}{9} = \frac{16}{x} = \frac{32}{36}\end{align*}

First, you can use two of the three ratios to solve for \begin{align*}x\end{align*}.

\begin{align*}\frac{8}{9} = \frac{16}{x}\end{align*}

Next, use algebra to solve the proportion.

\begin{align*}\begin{array}{rcl} \frac{8}{9} &=& \frac{16}{x} \\ 8x &=& 9 \times 16 \\ 8x &=& 144 \\ \frac{8x}{8} &=& \frac{144}{8} \\ x &=& 18 \end{array}\end{align*}

The answer is 18.

The value of \begin{align*}x\end{align*} is 18.

Review

Identify whether or not each pair of triangles is similar based on the ratios of their sides.

1. Triangle \begin{align*}A\end{align*} has side lengths of 2, 4, and 6. Triangle \begin{align*}B\end{align*} has side lengths of 6, 12 and 24. Are these triangles similar?

2. Triangle \begin{align*}C\end{align*} has side lengths of 4, 5, and 10. Triangle \begin{align*}B\end{align*} has side lengths of 2, 2.5 and 5. Are these two triangles similar?

3. Triangle \begin{align*}D\end{align*} has side lengths of 5, 8, and 12. Triangle \begin{align*}B\end{align*} has side lengths of 10, 16 and 24. Are these two triangles similar?

4. Triangle \begin{align*}A\end{align*} has side lengths of 10, 12, and 14. Triangle \begin{align*}B\end{align*} has side lengths of 5, 7 and 9. Are these two triangles similar?

5. Triangle \begin{align*}B\end{align*} has side lengths of 8, 14, and 20. Triangle \begin{align*}C\end{align*} has side lengths of 4, 7 and 10. Are these two triangles similar?

6. Triangle \begin{align*}E\end{align*} has side lengths of 20, 11 and 8. Triangle \begin{align*}F\end{align*} has side lengths of 10, 5.5 and 5. Are these two triangles similar?

7. Triangle \begin{align*}G\end{align*} has side lengths of 6, 8 and 12. Triangle \begin{align*}H\end{align*} has side lengths of 18, 24 and 36. Are these two triangles similar?

8. Triangle \begin{align*}I\end{align*} has side lengths of 8, 12, and 16. Triangle \begin{align*}J\end{align*} has side lengths of 4, 8 and 10. Are these two triangles similar?

Find the missing length by looking at each series of ratios. The top value represents the side lengths of the first similar triangle. The bottom value represents the side lengths of the second similar triangle.

9. \begin{align*}\frac{1}{2} = \frac{3}{6} = \frac{9}{x}\end{align*}

10. \begin{align*}\frac{3}{6} = \frac{6}{12} = \frac{10}{x}\end{align*}

11. \begin{align*}\frac{4}{2} = \frac{10}{x} = \frac{12}{6}\end{align*}

12. \begin{align*}\frac{6}{2} = \frac{9}{x} = \frac{12}{6}\end{align*}

13. \begin{align*}\frac{5}{10} = \frac{10}{20} = \frac{15}{x}\end{align*}

14. \begin{align*}\frac{12}{6} = \frac{20}{10} = \frac{15}{x}\end{align*}

15. \begin{align*}\frac{16}{x} = \frac{20}{5} = \frac{24}{6}\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 6.17. 

Vocabulary

AA Similarity Postulate

If two angles in one triangle are congruent to two angles in another triangle, then the two triangles are similar.

Congruent

Congruent figures are identical in size, shape and measure.

Indirect Measurement

Indirect measurement is the process of using the characteristics of similar triangles to measure distances.

Proportion

A proportion is an equation that shows two equivalent ratios.

Proportional Reasoning

Proportional reasoning involves deducing the relationship between the numerators or the denominators of a proportion. Anytime you have a proportion, there is some kind of relationship between the values.

Similar

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

Image Attributions

  1. [1]^ License: CC BY-NC 3.0
  2. [2]^ License: CC BY-NC 3.0
  3. [3]^ License: CC BY-NC 3.0
  4. [4]^ License: CC BY-NC 3.0
  5. [5]^ License: CC BY-NC 3.0

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