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# Indirect Measurement

## Measurement using similar triangles.

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Indirect Measurement

Have you ever tried to figure out the height of something very tall? Take a look at this dilemma.

The students working on the skate park design went to visit another skate park to help with their ideas. In the center of this skate park, there was a huge tall statue of a skate boarder.

"Look at that," Travis commented.

"Yes, that is very cool," Tania agreed.

"How tall do you think it is?" Travis asked.

"I don't know," Tania said.

“You can figure that out quite easily with math,” Mr. Henry their school advisor said overhearing the conversation.

“How can I do that?” Travis asked.

“How tall are you?”

“Five feet,” Travis said.

“Good and it looks like your shadow is about half as long as your height. Can you figure it out now?”

“Now, I have an idea,” he said smiling.

Do you have an idea what Travis is thinking about? If you pay attention to this Concept, you will know how to figure out the height of the statue.

### Guidance

We can use the properties of similar figures to measure things that are challenging to measure directly. We call this type of measurement indirect measurement.

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?

That is a good question! Think about a tree, it makes a shadow and the line from the end of the shadow to the top of the tree creates a triangle. It sounds confusing, but here is a diagram to help you.

Here is a palm tree. 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.

How would this work with the shadow of a person?

There is a triangle here too. Just because it is on an angle don’t let that fool you. It is still a triangle.

Alright, now let’s go back to the problem again.

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, we have to create two ratios. One will compare the heights of the man and the tree the other will compare the lengths of the shadows. Together, they will form a proportion because similar triangles are proportional and we 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 we can fill in the given information.

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

We are looking for the height of the tree, so that is where our variable goes. Now we can solve the proportion.

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

You can use similar triangles and proportions to measure difficult things. Indirect measurement makes the seemingly impossible, possible!!

Now it's time for you to try a few.

#### Example A

Are these triangles similar? Why or why not?

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

#### Example B

True or false. You need a proportion to effectively measure similar figure indirectly.

Solution: True

#### Example C

True or false. You can use cross - products or equal ratios to measure indirectly.

Solution: True

Here is the original problem once again.

The students working on the skate park design went to visit another skate park to help with their ideas. In the center of this skate park, there was a huge tall statue of a skate boarder.

"Look at that," Travis commented.

"Yes, that is very cool," Tania agreed.

"How tall do you think it is?" Travis asked.

"I don't know," Tania said.

“You can figure that out quite easily with math,” Mr. Henry their school advisor said overhearing the conversation.

“How can I do that?” Travis asked.

“How tall are you?”

“Five feet,” Travis said.

“Good and it looks like your shadow is about half as long as your height. Can you figure it out now?”

“Now, I have an idea,” he said smiling.

Let’s think about how Travis and Tania could figure out the height of the statue. We know that Travis is five feet tall and that his shadow is half as long as he is tall. Now we can write a ratio to compare Travis' height to his shadow’s length.

\begin{align*}\frac{Travis' \ height}{Shadow' s \ length} = \frac{5 \ feet}{2.5 \ feet}\end{align*}

Next, we figure out the height of the statue. Travis and Tania figure out very quickly that they need to figure out the length of the shadow of the statue to figure out the height of the statue. Once they know the length of the shadow, they can use proportional reasoning and indirect measurement to figure out the statue’s height.

Approximating 1 foot using a length a little longer than Travis' sneaker, they measure \begin{align*}32 \frac{1}{2}\end{align*} feet. It is not an exact measure, but they feel that it is very close.

Now they wrote the following proportion.

\begin{align*}\frac{5 \ ft}{2.5 \ ft} = \frac{x}{32.5 \ ft}\end{align*}

Taking out a notebook, Tania cross multiplies to solve the proportion.

\begin{align*}5(32.5) &= 2.5x\\ 162.5 &= 2.5x\\ x &= 65\end{align*}

The sculpture is approximately 65 feet tall.

### Vocabulary

Congruent
having the same size and shape and measurement
Similar
having the same shape, but not the same size. Similar shapes are proportional to each other.
Corresponding
matching-corresponding sides between two triangles are sides that match up
Ratio
a way of comparing two quantities
Proportion
a pair of equal ratios.
Indirect Measurement
using the characteristics of similar triangles to measure challenging things or distances.

### Guided Practice

Here is one for you to try on your own.

What is the missing side length?

Look at these two rectangles. First, look and see if we can figure out the relationship between the two figures. To do this, we compare the side lengths of each part of the two figures.

We need to figure out the measurement of side \begin{align*}GH\end{align*} in the second rectangle.

You can see that the measurements in the second rectangle are half as big as the measurements in the first. Also, you know that the opposite sides of a rectangle are congruent. Therefore, the missing side length is 4.

### Practice

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

Directions: 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 \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*}?

11. 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*}?

Directions: Solve each proportion for the missing side length.

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

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

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

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

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### 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.