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# Indirect Measurement

## Measurement using similar triangles.

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Indirect Measurement

What if you wanted to measure the height of a flagpole using your friend George? He is 6 feet tall and his shadow is 10 feet long. At the same time, the shadow of the flagpole was 85 feet long. How tall is the flagpole?

### Indirect Measurement

An application of similar triangles is to measure lengths indirectly. The length to be measured would be some feature that was not easily accessible to a person, such as the width of a river or canyon and the height of a tall object. To measure something indirectly, you need to set up a pair of similar triangles.

#### Measuring Something Indirectly

1. A tree outside Ellie’s building casts a 125 foot shadow. At the same time of day, Ellie casts a 5.5 foot shadow. If Ellie is 4 feet 10 inches tall, how tall is the tree?

Draw a picture. From the picture to the right, we see that the tree and Ellie are parallel, therefore the two triangles are similar to each other. Write a proportion.

4ft,10inxft=5.5ft125ft\begin{align*}\frac{4ft, 10in}{xft}=\frac{5.5ft}{125ft}\end{align*}

Notice that our measurements are not all in the same units. Change both numerators to inches and then we can cross multiply.

58inxft=66in125ft58(125)7250x=66(x)=66x109.85 ft\begin{align*}\frac{58in}{xft}=\frac{66in}{125ft} \longrightarrow 58(125) &= 66(x)\\ 7250 &= 66x\\ x & \approx 109.85 \ ft\end{align*}

2. Cameron is 5 ft tall and casts a 12 ft shadow. At the same time of day, a nearby building casts a 78 ft shadow. How tall is the building?

To solve, set up a proportion that compares height to shadow length for Cameron and the building. Then solve the equation to find the height of the building. Let x\begin{align*}x\end{align*} represent the height of the building.

5ft12ft12xx=x78ft=390=32.5ft\begin{align*} \frac{5 ft}{12 ft}&=\frac{x}{78 ft} \\ 12x&=390 \\ x&=32.5 ft\end{align*}

The building is 32.5\begin{align*}32.5\end{align*} feet tall.

3. The Empire State Building is 1250 ft. tall. At 3:00, Pablo stands next to the building and has an 8 ft. shadow. If he is 6 ft tall, how long is the Empire State Building’s shadow at 3:00?

Similar to Example B, solve by setting up a proportion that compares height to shadow length. Then solve the equation to find the length of the shadow. Let x\begin{align*}x\end{align*} represent the length of the shadow.

6ft8ft6xx=1250ftx=10000=1666.67ft\begin{align*}\frac{6 ft}{8 ft}&=\frac{1250 ft}{x}\\ 6x&=10000\\ x&=1666.67 ft\end{align*}

The shadow is approximately 1666.67\begin{align*}1666.67\end{align*} feet long.

#### Flagpole Problem Revisited

It is safe to assume that George and the flagpole stand vertically, making right angles with the ground. Also, the angle where the sun’s rays hit the ground is the same for both. The two trianglesare similar. Set up a proportion.

1085=6x10xx=510=51 ft.\begin{align*}\frac{10}{85} = \frac{6}{x} \longrightarrow 10x &= 510\\ x &= 51 \ ft.\end{align*}

The height of the flagpole is 51 feet.

### Examples

In order to estimate the width of a river, the following technique can be used. Use the diagram.

Place three markers, O,C,\begin{align*}O, C,\end{align*} and E\begin{align*}E\end{align*} on the upper bank of the river. E\begin{align*}E\end{align*} is on the edge of the river and OC¯¯¯¯¯¯¯¯CE¯¯¯¯¯¯¯¯\begin{align*}\overline {OC} \perp \overline{CE}\end{align*}. Go across the river and place a marker, N\begin{align*}N\end{align*} so that it is collinear with C\begin{align*}C\end{align*} and E\begin{align*}E\end{align*}. Then, walk along the lower bank of the river and place marker A\begin{align*}A\end{align*}, so that CN¯¯¯¯¯¯¯¯NA¯¯¯¯¯¯¯¯\begin{align*}\overline{CN} \perp \overline{NA}\end{align*}. OC=50\begin{align*}OC = 50\end{align*} feet, CE=30\begin{align*}CE = 30\end{align*} feet, NA=80\begin{align*}NA = 80\end{align*} feet.

#### Example 1

Is \begin{align*}\triangle OCE \sim \triangle ANE?\end{align*} How do you know?

Yes. \begin{align*}\angle{C} \cong \angle{N}\end{align*} because they are both right angles. \begin{align*}\angle{OEC} \cong \angle{AEN}\end{align*} because they are vertical angles. This means \begin{align*}\triangle OCE \sim \triangle ANE\end{align*} by the AA Similarity Postulate.

#### Example 2

Is \begin{align*}\overline {OC} \| \overline {NA}?\end{align*} How do you know?

Since the two triangles are similar, we must have \begin{align*}\angle{EOC} \cong \angle{EAN}\end{align*}. These are alternate interior angles. When alternate interior angles are congruent then lines are parallel, so \begin{align*}\overline {OC} \| \overline {NA}\end{align*}.

#### Example 3

What is the width of the river? Find \begin{align*}EN\end{align*}.

Set up a proportion and solve by cross-multiplying.

\begin{align*}\frac{30 ft}{EN}&=\frac{50 ft}{80 ft}\\ 50(EN)&=2400 \\ EN&=48\end{align*}

The river is \begin{align*}48\end{align*} feet wide.

### Review

The technique from the guided practice section was used to measure the distance across the Grand Canyon. Use the picture below and \begin{align*}OC = 72 \ ft , CE = 65 \ ft\end{align*}, and \begin{align*}NA = 14,400 \ ft\end{align*} for problems 1 - 3.

1. Find \begin{align*}EN\end{align*} (the distance across the Grand Canyon).
2. Find \begin{align*}OE\end{align*}.
3. Find \begin{align*}EA\end{align*}.
1. Janet wants to measure the height of her apartment building. She places a pocket mirror on the ground 20 ft from the building and steps backwards until she can see the top of the build in the mirror. She is 18 in from the mirror and her eyes are 5 ft 3 in above the ground. The angle formed by her line of sight and the ground is congruent to the angle formed by the reflection of the building and the ground. You may wish to draw a diagram to illustrate this problem. How tall is the building?
2. Sebastian is curious to know how tall the announcer’s box is on his school’s football field. On a sunny day he measures the shadow of the box to be 45 ft and his own shadow is 9 ft. Sebastian is 5 ft 10 in tall. Find the height of the box.
3. Juanita wonders how tall the mast of a ship she spots in the harbor is. The deck of the ship is the same height as the pier on which she is standing. The shadow of the mast is on the pier and she measures it to be 18 ft long. Juanita is 5 ft 4 in tall and her shadow is 4 ft long. How tall is the ship’s mast?
4. Evan is 6 ft tall and casts a 15 ft shadow. A the same time of day, a nearby building casts a 30 ft shadow. How tall is the building?
5. Priya and Meera are standing next to each other. Priya casts a 10 ft shadow and Meera casts an 8 ft shadow. Who is taller? How do you know?
6. Billy is 5 ft 9 inches tall and Bobby is 6 ft tall. Bobby's shadow is 13 feet long. How long is Billy's shadow?
7. Sally and her little brother are walking to school. Sally is 4 ft tall and has a shadow that is 3 ft long. Her little brother's shadow is 2 ft long. How tall is her little brother?
8. Ray is outside playing basketball. He is 5 ft tall and at this time of day is casting a 12 ft shadow. The basketball hoop is 10 ft tall. How long is the basketball hoop's shadow?
9. Jack is standing next to a very tall tree and wonders just how tall it is. He knows that he is 6 ft tall and at this moment his shadow is 8 ft long. He measures the shadow of the tree and finds it is 90 ft. How tall is the tree?
10. Jason, who is 4 ft 9 inches tall is casting a 6 ft shadow. A nearby building is casting a 42 ft shadow. How tall is the building?
11. Alexandra, who is 5 ft 8 in tall is casting a 12 ft shadow. A nearby lamppost is casting a 20 ft shadow. How tall is the lamppost?
12. Use shadows or a mirror to measure the height of an object in your yard or on the school grounds. Draw a picture to illustrate your method.

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### Vocabulary Language: English

TermDefinition
AA Similarity Postulate If two angles in one triangle are congruent to two angles in another triangle, then the two triangles are similar.
Proportion A proportion is an equation that shows two equivalent ratios.