<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />

# Indirect Measurement

## Measurement using similar triangles.

Estimated6 minsto complete
%
Progress
Practice Indirect Measurement
Progress
Estimated6 minsto complete
%
Indirect Measurement

### Let's Think About It

Credit: Tim (Timothy) Pearce
Source: https://www.flickr.com/photos/timpearcelosgatos/3499121180
License: CC BY-NC 3.0

Franz has been given the honor of hoisting the flag at his school. He stands next to the flagpole while he hooks the flag. His classmates notice two shadows. Franz is 4 feet tall and his shadow ends at the bench that is 6 feet away from him. The flagpole is 25 feet tall, but the students don't have a direct way to measure the distance of its shadow. Franz's teacher suggests that the students use indirect measurement to calculate the length of the flagpole's shadow.

Height of FranzHeight of flagpole=Shadow length of FranzShadow length of flagpole\begin{align*}\frac{Height\ of\ Franz}{Height\ of\ flagpole} = \frac{Shadow\ length\ of\ Franz}{Shadow\ length\ of\ flagpole}\end{align*}

425=6x\begin{align*}\frac{4'}{25'} = \frac{6'}{x}\end{align*}

How can the students determine the length of the flagpole's shadow?

In this concept, you will learn how to make an indirect measurement.

### Guidance

You can use the properties of similar figures to measure things that are challenging to measure directly. This type of measurement is called indirect measurement.

Let's look at an example.

Jamie’s Dad is six feet tall. Standing outside, his shadow is eight feet long. A tree is next to him. The tree has a shadow that is sixteen feet long. Given these dimensions, how tall is the tree?

Here is a diagram to help you.

You can see from the picture that the tree itself is one side of the triangle, that the shadow length is another side of the triangle and that the diagonal from the top of the tree to the top of the shadow forms the hypotenuse (the longest side) of the triangle.

Let's compare it to the shadow of a person.

There is a triangle here, too.

Now let's combine this information.

Jamie’s dad is six feet tall. Standing outside, his shadow is eight feet long. A tree is next to him. The tree has a shadow that is sixteen feet long. Given these dimensions, how tall is the tree?

To solve this, you have to create two ratios. One will compare the heights of the man and the tree and the other will compare the lengths of the shadows. Together, they will form a proportion because similar triangles are proportional and you have already seen how triangles are created with people or things and shadows.

Height of ManHeight of Tree=Shadow length of manShadow length of tree\begin{align*}\frac{Height\ of\ Man}{Height\ of\ Tree} = \frac{Shadow\ length\ of\ man}{Shadow\ length\ of\ tree}\end{align*}

Now, fill in the given information.

6x=816\begin{align*}\frac{6'}{x} = \frac{8'}{16'}\end{align*}

You are looking for the height of the tree, so that is where the variable goes.

Now identify the relationship between the numerical values in the numerator and the denominator.

The denominator is twice the numerator.

Write and solve an equation to calculate the value of x\begin{align*}x\end{align*}.

x=6×2=12\begin{align*}x = 6 \times 2=12\end{align*}

The answer is 12 feet. The tree is 12 feet tall.

### Guided Practice

What is the missing side length?



First, figure out the relationship between the two figures.

6"3"=8"x\begin{align*}\frac{6"}{3"}=\frac{8"}{x}\end{align*}

Next, find the cross products.

6x=8×3\begin{align*}6x = 8\times 3\end{align*}

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

x=8×36=246=4\begin{align*}x=\frac{8\times 3}{6}=\frac{24}{6}=4\end{align*}

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

### Examples

#### Example 1

Are these triangles similar?

First, check to see if the two triangles are the same size.

No

Next, check to see if the corresponding angles are congruent.

Yes

Then, determine if the two triangles are similar.

Yes

The answer is that the two triangles are similar.

#### Example 2

Solve for the missing value.

46=12x\begin{align*}\frac{4'}{6'}=\frac{12'}{x}\end{align*}

First, use cross products.

4x=6×12\begin{align*}4x=6\times 12\end{align*}

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

x=(6×12)4=724=18\begin{align*}x=\frac{(6\times 12)}{4}=\frac{72}{4}=18\end{align*}

The answer is that x=18\begin{align*}x=18\end{align*}.

#### Example 3

Use the proportion to indirectly solve for the length of the woman's shadow.

Height of WomanHeight of Tree=Shadow length of womanShadow length of tree\begin{align*}\frac{Height\ of\ Woman}{Height\ of\ Tree} = \frac{Shadow\ length\ of\ woman}{Shadow\ length\ of\ tree}\end{align*}

515=x30\begin{align*}\frac{5'}{15'} = \frac{x}{30'}\end{align*}

First, use cross products.

15x=5×30\begin{align*}15x=5\times 30\end{align*}

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

x=(5×30)15=15015=10\begin{align*}x=\frac{(5\times 30)}{15}=\frac{150}{15}=10\end{align*}

The answer is that x=10\begin{align*}x=10\end{align*}. The woman's shadow is 10 feet long.

### Follow Up

Credit: Tim (Timothy) Pearce
Source: https://www.flickr.com/photos/timpearcelosgatos/3499121180
License: CC BY-NC 3.0

Remember Franz and the flagpole? Franz is 4 feet tall and his shadow ends at the bench that is 6 feet away from him. Franz's teacher suggests that the students use indirect measurement to calculate the length of the flagpole's shadow.

Height of FranzHeight of flagpole=Shadow length of FranzShadow length of flagpole\begin{align*}\frac{Height\ of\ Franz}{Height\ of\ flagpole} = \frac{Shadow\ length\ of\ Franz}{Shadow\ length\ of\ flagpole}\end{align*}

425=6x\begin{align*}\frac{4'}{25'} = \frac{6'}{x}\end{align*}

What is the length of the flagpole's shadow?

First, find the cross products.

4x=6×25\begin{align*}4x=6\times 25\end{align*}

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

x=6×254=1504=37.5\begin{align*}x=\frac{6\times 25}{4}=\frac{150}{4}=37.5\end{align*}

The answer is that the flagpole's shadow is 37.5 feet long.

### Explore More

Use indirect measurement and proportions to determine if the figures are similar.

Use what you have learned about similar triangles and indirect measurement to solve each of the following problems.

5. If a person who is five feet tall casts a shadow that is 8 feet long, how tall is a building that casts a shadow that is 24 feet long?

6. If a tree stump that is two feet tall casts a shadow that is one foot long, how long is the shadow of a tree that is ten feet at the same time of day?

7. If a 6 foot pole has a shadow that is eight feet long, how tall is a nearby tower that has a shadow that is 16 feet long?

8. If a lifeguard tower is ten feet tall and casts a shadow that is eight feet long, how tall is a person who casts a shadow that is four feet long?

9. Draw the triangle in on the following picture.

10. Side m\begin{align*}m\end{align*} in triangle LMN\begin{align*}LMN\end{align*} corresponds to side c\begin{align*}c\end{align*} in the smaller triangle BCD\begin{align*}BCD\end{align*}. Side m\begin{align*}m\end{align*} is 12 cm long and the scale factor is 4. What is the measure of side c\begin{align*}c\end{align*}?

11. Side q\begin{align*}q\end{align*} in triangle PQR\begin{align*}PQR\end{align*} corresponds to side y\begin{align*}y\end{align*} in the smaller triangle XYZ\begin{align*}XYZ\end{align*}. Side y\begin{align*}y\end{align*} is 8 inches long and the scale factor is 7. What is the measure of side q\begin{align*}q\end{align*}?

Solve each proportion for the missing side length.

12. 34=x12\begin{align*}\frac{3}{4} = \frac{x}{12}\end{align*}

13. 36=1x\begin{align*}\frac{3}{6} = \frac{1}{x}\end{align*}

14. 58=1x\begin{align*}\frac{5}{8} = \frac{1}{x}\end{align*}

15. 710=x30\begin{align*}\frac{7}{10} = \frac{x}{30}\end{align*}

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

Color Highlighted Text Notes
Please to create your own Highlights / Notes
Show More

### Vocabulary Language: English

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.

Corresponding

The corresponding sides between two triangles are sides in the same relative position.

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.

Similar

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

### Image Attributions

1. [1]^ Credit: Tim (Timothy) Pearce; Source: https://www.flickr.com/photos/timpearcelosgatos/3499121180; License: CC BY-NC 3.0
2. [2]^ Credit: Tim (Timothy) Pearce; Source: https://www.flickr.com/photos/timpearcelosgatos/3499121180; License: CC BY-NC 3.0

### Explore More

Sign in to explore more, including practice questions and solutions for Indirect Measurement.
Please wait...
Please wait...