Suppose an architectural firm is constructing a building, and they first build a model that is 4 feet tall. If the scale is 1:125 when compared to the height of the real building, can you figure out how tall the real building is? What proportion would you set up?

### Scale and Indirect Measurements

We are occasionally faced with having to make measurements of things that would be difficult to measure directly: the height of a tall tree, the width of a wide river, the height of the moon’s craters, and even the distance between two cities separated by mountainous terrain. In such circumstances, measurements can be made **indirectly**, using proportions and similar triangles. Such indirect methods link measurement with geometry and numbers.

#### Scale

A map is a two-dimensional, geometrically accurate representation of a section of the Earth’s surface. Maps are used to show, pictorially, how various geographical features are arranged in a particular area. The **scale** of the map describes the relationship between distances on the map and the corresponding distances on the Earth's surface. These measurements are expressed as a fraction or a ratio.

Recall that there are different ways to write a ratio: using the fraction bar, using a colon, and in words. Outside of mathematics books, ratios are often written as two numbers separated by a colon (:). Here is a table that compares ratios written in two different ways.

Ratio |
Is Read As |
Equivalent To |
---|---|---|

1:20 | one to twenty | \begin{align*}\left (\frac{1}{20} \right )\end{align*} |

2:3 | two to three | \begin{align*}\left (\frac{2}{3} \right )\end{align*} |

1:1000 | one to one-thousand | \begin{align*}\left (\frac{1}{1000} \right )\end{align*} |

If a map had a scale of 1:1000 (“one to one-thousand”), one unit of measurement on the map (1 inch or 1 centimeter, for example) would represent 1000 of the same units on the ground. A 1:1 (one to one) map would be a map as large as the area it shows!

#### Let's solve the following problems using proportions:

- Anne is visiting a friend in London and is using the map above to navigate from Fleet Street to Borough Road. She is using a 1:100,000 scale map, where 1 cm on the map represents 1 km in real life. Using a ruler, she measures the distance on the map as 8.8 cm. How far is the real distance from the start of her journey to the end?

The scale is the ratio of distance on the map to the corresponding distance in real life and can be written as a proportion.

\begin{align*}\frac{\text{dist.on map}}{\text{real dist.}} = \frac{1}{100, 000}\end{align*}

By substituting known values, the proportion becomes:

\begin{align*}\frac{8.8 \ cm}{\text{real dist.} (x)} & = \frac{1}{100,000} && \text{Cross multiply}. \\ 880000 \ cm & = x && 100 \ cm = 1\ m. \\ x & = 8800 \ m && 1000 \ m = 1\ km.\end{align*}

The distance from Fleet Street to Borough Road is \begin{align*}8800\ m\end{align*} or \begin{align*}8.8\ km\end{align*}.

We could, in this case, use our intuition: the \begin{align*}1 \ cm = 1 \ km\end{align*} scale indicates that we could simply use our reading in centimeters to give us our reading in km. Not all maps have a scale this simple. In general, you will need to refer to the map scale to convert between measurements on the map and distances in real life!

- Oscar is trying to make a scale drawing of the Titanic, which he knows was 883 feet long. He would like his drawing to be 1:500 scale. How long, in inches, must his sheet of paper be?

We can reason that since the scale is 1:500 that the paper must be \begin{align*}\frac{883}{500} = 1.766\ feet\end{align*} long. Converting to inches gives the length as \begin{align*}12 (1.766) \ in = 21.192 \ in\end{align*}.

The paper should be at least 22 inches long.

#### Indirect Measurement

Not everything has a scale. Architecture such as the St. Louis Arch, St. Basil’s Cathedral, or the Eiffel Tower does not have a scale written on the side. It may be necessary to measure such buildings. To do so requires knowledge of similar figures and a method called **indirect measurement**.

**Similar figures** are often used to make indirect measurements. Two shapes are said to be **similar** if they are the same shape and “in proportion.” The ratio of every measurable length in one figure to the corresponding length in the other is the same. **Similar triangles** are often used in indirect measurement.

#### Let's solve the following problem with indirect measurement:

Anatole is visiting Paris, and wants to know the height of the Eiffel Tower. Since he's unable to speak French, he decides to measure it in three steps.

- He measures out a point 500 meters from the base of the tower, and places a small mirror flat on the ground.
- He stands behind the mirror in such a spot that standing upright he sees the top of the tower reflected in the mirror.
- He measures both the distance from the spot where he stands to the mirror (2.75 meters) and the height of his eyes from the ground (1.8 meters).

Explain how Anatole is able to determine the height of the Eiffel Tower from these numbers and determine what that height is.

First, we will draw and label a scale diagram of the situation.

The Law of Reflection states, “The angle at which the light reflects off the mirror is the same as the angle at which it hits the mirror.” Using this principle and the figure above, you can conclude that these triangles are similar with proportional sides.

This means that the ratio of the long leg in the large triangle to the length of the long leg in the small triangle is the **same ratio** as the length of the short leg in the large triangle to the length of the short leg in the small triangle.

\begin{align*}\frac{500 \ m}{2.75 \ m} & = \frac{x}{1.8 \ m} \\ 1.8 \cdot \frac{500}{2.75} & = \frac{x}{1.8} \cdot 1.8 \\ 327.3 & \approx x\end{align*}

The Eiffel Tower, according to this calculation, is approximately 327.3 meters high.

### Examples

#### Example 1

Earlier, you were told that a model of a building is 4 feet tall. The scale is 1:125 when compared to the height of the real building. How tall is the real building?

Let's set the scale up as a ratio:

\begin{align*}\frac{\text{model height}}{\text{real height}} = \frac{1}{125}\end{align*}Now, substitute in known values.

\begin{align*}\frac{\text{model height}}{\text{real height}} = \frac{1}{125}\\ \frac{4}{\text{real height}} = \frac{1}{125}\end{align*}

To make it easier to read the equation, let's call the real height of the building the variable \begin{align*}h\end{align*}.

To solve, cross multiply:

\begin{align*}\frac{4}{h}=\frac{1}{125}\\ 4\times 125 = 1\times h\\ 500 = h\end{align*}

The real building is 500 feet tall.

#### Example 2

Bernard is looking at a lighthouse and wondering how high it is. He notices that it casts a long shadow, which he measures at 200 meters long. At the same time, he measures his own shadow at 3.1 meters long. Bernard is 1.9 meters tall. How tall is the lighthouse?

Begin by drawing a scale diagram.

You can see there are two right triangles. The angle at which the sun causes the shadow from the lighthouse to fall is the same angle at which Bernard’s shadow falls. We have two similar triangles, so we can again say that the ratio of the corresponding sides is the same.

\begin{align*}\frac{200 \ m}{3.1 \ m} & = \frac{x}{1.9 \ m} \\ 1.9 \cdot \frac{200 \ m}{3.1 \ m} & = \frac{x}{1.9 \ m} \cdot 1.9 \\ 122.6 & \approx x\end{align*}

The lighthouse is approximately 122.6 meters tall.

### Review

- Define similar figures.
- What is true about similar figures?
- What is the process of indirect measurement? When is indirect measurement particularly useful?
- State the Law of Reflection. How does this law relate to similar figures?
- A map has a 1 inch : 20 mile scale. If two cities are 1,214 miles apart, how far apart would they be on this map?
- What would a scale of 1 mile : 1 mile mean on a map? What problems would the cartographer encounter?
- A woman 66 inches tall stands beside a tree. The length of her shadow is 34 inches and the length of the tree’s shadow is 98 inches. To the nearest foot, how tall is the tree?

- Use the scale diagram above to determine:
- The length of the helicopter (cabin to tail)
- The height of the helicopter (floor to rotors)
- The length of one main rotor
- The width of the cabin
- The diameter of the rear rotor system

- On a sunny morning, the shadow of the Empire State Building is 600 feet long. At the same time, the shadow of a yardstick (3 feet long) is 1 foot, \begin{align*}5 \frac{1}{4}\end{align*} inches. How high is the Empire State building?
- Omar and Fredrickson are 12.4 inches apart on a map with a scale of 1.2 inches : 15 miles. How far apart are their two cities in miles?
- A man 6 feet tall is standing next to a dog. The man’s shadow is 9 feet and the dog’s shadow is 6 feet. How tall is the dog?
- A model house is 12 inches wide. It was built with a ratio of 3 inches : 4 meters. How wide is the actual house?
- Using the diagram below and assuming the two triangles are similar, find \begin{align*}\overline{DE}\end{align*}, the length of segment \begin{align*}DE\end{align*}.
- A 42.9-foot flagpole casts a 253.1-foot shadow. What is the length of the shadow of a woman 5 feet 5 inches standing next to the flagpole?

**Mixed Review**

- Solve for \begin{align*}a: \ -(7-7a)+4a=-23+3a\end{align*}.
- How is evaluating different from solving? Provide an example to help you illustrate your explanation.
- Simplify \begin{align*}\sqrt{243}\end{align*}.
- Simplify: \begin{align*}2(8g+2)-1+4g-(2-5g)\end{align*}.
- Draw a graph that is not a function. Explain why your picture is not a function.
- Jose has \begin{align*}\frac{2}{3}\end{align*} the amount of money that Chloe has. Chloe has four dollars less than Huey. Huey has $26. How much does Jose have?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 3.8.