In the 1800s, the average house was \begin{align*}20 \times 40 \text{ feet}\end{align*}. So the average area was 800 square feet. In order to create a model of a log cabin this size, you are going to use a scale factor of \begin{align*}\frac{1}{16}\end{align*}. What will be the dimensions of the model?

In this concept, you will learn to recognize dilations.

### Dilations

There are all kinds of transformations. You can flip or reflect a figure, translate or slide a figure and rotate a figure. You can also stretch or shrink a figure to create a new one. This is called a **dilation**. A **dilation** is a transformation created by a scale factor.

You can create a dilation that is smaller or larger than the original figure. Either way, a similar figure is created through a dilation.

Let’s think about scale factors.

The **scale factor** is the ratio that determines the proportional relationship between the sides of similar figures.

For the pairs of sides to be proportional to each other, they must have the same scale factor. In other words, similar figures have congruent angles and sides with the same scale factor. With a scale factor of two, each side of the larger figure is exactly twice as long as the corresponding side in the smaller figure.

Let’s look at an example.

A figure has a side length of 3 feet. What would be the corresponding side length of the next figure if the scale factor were 4?

You know the length of one of the sides of the first figure and you know the scale factor. To figure out the new length, you can multiply the scale factor times the first length.

\begin{align*}3 \times 4=12\end{align*}

The answer is 12.

The length of the corresponding side of the second figure is 12 feet.

When you have a figure that is larger than the original, you have a scale factor that is greater than one. If you have a figure that is smaller than the original, then you have a scale factor that is less than one or a fraction.

Let’s look at an example.

A figure has a side length of 5 meters. What would be the corresponding side length of the new figure if the scale factor is \begin{align*}\frac{1}{2}\end{align*}?

To figure this out, you have to take the given length of the first figure and multiply it by one-half. Remember multiplying by one-half is the same as dividing by 2. This will give you the corresponding length of the second figure.

\begin{align*}5 \times \frac{1}{2}=\frac{5}{2}\end{align*}

The answer is \begin{align*}\frac{5}{2}\end{align*}.

The length of the corresponding side of the new figure is \begin{align*}\frac{5}{2}\end{align*} or 2.5 meters.

Now that you understand dilations, you should be able to work with them on the coordinate plane. Once again, you will be using coordinate notation to describe the different dilations that are created on the coordinate plane.

Let’s look at this figure and then see how you can graph the dilation of it.

Graphing dilations of geometric figures is actually fairly easy to do when you know the scale factor. You simply multiply both coordinates for each vertex by the scale factor to produce new coordinates.

Suppose you want to make an enlargement of the rectangle above using a scale factor of 3. You need to multiply each coordinate by 3.

\begin{align*}\begin{array}{rcl} (-2, -3) \times 3 &=& (-6, -9) \\ (-2, 3) \times 3 &=& (-6, 9) \\ (2, 3) \times 3 &=& (6, 9) \\ (2, -3) \times 3 &=& (6, -9) \end{array}\end{align*}

Now you can graph each of these new points on the coordinate plane.

You can also create a reduction. You create a reduction by dividing each coordinate by the scale factor. This will give you the new measurements of the figure.

### Examples

#### Example 1

Earlier, you were given a problem about the log cabin model.

To solve this problem, you begin with the actual dimensions of the log cabin. The log cabin has real world dimensions of \begin{align*}20 \times 40 \text{ feet}\end{align*}.

You are using a scale factor of \begin{align*}\frac{1}{16}\end{align*}. That means that the dilation will be a reduction. You need to multiply both dimensions by \begin{align*}\frac{1}{16}\end{align*}.

The answers are 1.25 and 2.5.

The scale model will be 1.25 feet wide and 2.5 feet long.

#### Example 2

Graph a reduction of the following figure if the scale factor is \begin{align*}\frac{1}{2}\end{align*}.

First, find the coordinates of the original figure.

\begin{align*}(-6, -2)\end{align*}

Next, using the scale factor of \begin{align*}\frac{1}{2}\end{align*}, find the coordinated of the dilated shape. Remember that multiplying by \begin{align*}\frac{1}{2}\end{align*} is the same as dividing by 2.

\begin{align*}\begin{array}{rcl} (2, 4) \div 2 &=& (1, 2) \\ (8, -4) \div 2 &=& (4, -2) \\ (-6, -2) \div 2 &=& (-3, -1) \end{array}\end{align*}

Then, graph the new dilated shape.

#### Example 3

A quadrilateral with side measures of 6, 15, 27, and 30 is to be dilated using a scale factor of \begin{align*}\frac{1}{3}\end{align*}.

Multiply each side measure by \begin{align*}\frac{1}{3}\end{align*}.

\begin{align*}\begin{array}{rcl} 6 \times \frac{1}{3} &=& 2 \\ \\ 15 \times \frac{1}{3} &=& 5 \\ \\ 27 \times \frac{1}{3} &=& 9 \\ \\ 30 \times \frac{1}{3} &=& 10 \end{array}\end{align*}

The answer is 2, 5, 9, and 10.

The new side measures of the dilated quadrilateral are 2, 5, 9, and 10 respectively.

#### Example 4

A quadrilateral with side measures of 6, 15, 27, and 30 is to be dilated using a scale factor of \begin{align*}\frac{1}{2}\end{align*}.

Multiply each side measure by \begin{align*}\frac{1}{2}\end{align*}.

The answer is 3, 7.5, 13.5, and 15.

The new measures of the dilated sides of the quadrilateral are 3, 7.5, 13.5 and 15 respectively.

#### Example 5

A quadrilateral with side measures of 6, 15, 27, and 30 is to be dilated using a scale factor of 2.

Multiply each side measure by 2.

\begin{align*}\begin{array}{rcl} 6 \times 2 &=& 12 \\ 15 \times 2 &=& 20 \\ 27 \times 2 &=& 54 \\ 30 \times 2 &=& 60 \end{array}\end{align*}

The answer is 12, 20, 54, and 60.

The new side lengths of the dilated quadrilateral are 12, 20, 54, and 60 respectively.

### Review

Use each scale factor to determine the new dimensions of each figure.

- A triangle with side measures of 4, 5, 9 and a scale factor of 2.
- A triangle with side measures of 4, 5, 9 and a scale factor of 3.
- A triangle with side measures of 4, 5, 9 and a scale factor of 4.
- A triangle with side measures of 8, 10, 14 and a scale factor of 2.
- A triangle with side measures of 8, 10, 14 and a scale factor of 4.
- A triangle with side measures of 2, 4, 6 and a scale factor of 2.
- A quadrilateral with side measures of 4, 6, 8, 10 and a scale factor of \begin{align*}\frac{1}{2}\end{align*}.
- A quadrilateral with side measures of 12, 16, 20, 24 and a scale factor of \begin{align*}\frac{1}{4}\end{align*}.
- A quadrilateral with side measures of 4, 6, 8, 10 and a scale factor of 2.
- A quadrilateral with side measures of 4, 6, 8, 10 and a scale factor of 3.
- A quadrilateral with side measures of 4, 6, 8, 10 and a scale factor of 4.
- A quadrilateral with side measures of 9, 12, 18, 24 and a scale factor of \begin{align*}\frac{1}{3}\end{align*}.
- A quadrilateral with side measures of 9, 12, 18, 24 and a scale factor of 2.
- A quadrilateral with side measures of 9, 12, 18, 24 and a scale factor of 3.
- A quadrilateral with side measures of 8, 12, 16, 24 and a scale factor of .
- A quadrilateral with side measures of 9, 12, 18, 24 and a scale factor of .

### Review (Answers)

To see the Review answers, open this PDF file and look for section 6.18.