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Indirect Measurement

Measurement using similar triangles.

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Indirect Measurement
Source: https://www.flickr.com/photos/timpearcelosgatos/3499121180
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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.

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

 \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.

Indirect Measurement

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.

\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.

\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 \begin{align*}x\end{align*}.

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

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

Examples

Example 1

Earlier, you were given a problem about 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.

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

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

What is the length of the flagpole's shadow?

First, find the cross products.

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

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

 \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.

Example 2

What is the missing side length?

First, figure out the relationship between the two figures.

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

Next, find the cross products.

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

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

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

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

Example 3

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 4

Solve for the missing value.

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

First, use cross products.

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

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

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

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

Example 5

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

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

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

First, use cross products.

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

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

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

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

Review

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.

  1. 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?
  2. 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?
  3. 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?
  4. 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?
  5. Draw the triangle in on the following picture.

  1. Side \begin{align*}m\end{align*} in triangle \begin{align*}LMN\end{align*} corresponds to side \begin{align*}c\end{align*} in the smaller triangle \begin{align*}BCD\end{align*}. Side \begin{align*}m\end{align*} is 12 cm long and the scale factor is 4. What is the measure of side \begin{align*}c\end{align*}?
  2. Side \begin{align*}q\end{align*} in triangle \begin{align*}PQR\end{align*} corresponds to side \begin{align*}y\end{align*} in the smaller triangle \begin{align*}XYZ\end{align*}. Side \begin{align*}y\end{align*} is 8 inches long and the scale factor is 7. What is the measure of side \begin{align*}q\end{align*}?

Solve each proportion for the missing side length.

  1. \begin{align*}\frac{3}{4} = \frac{x}{12}\end{align*}
  2. \begin{align*}\frac{3}{6} = \frac{1}{x}\end{align*}
  3. \begin{align*}\frac{5}{8} = \frac{1}{x}\end{align*}
  4. \begin{align*}\frac{7}{10} = \frac{x}{30}\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 9.19. 

Resources

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