Which one of the following figures represents a dilation? Explain.

### Dilations

In geometry, a transformation is an operation that moves, flips, or changes a shape to create a new shape. A dilation is a type of transformation that enlarges or reduces a figure (called the preimage) to create a new figure (called the image). The scale factor, r, determines how much bigger or smaller the dilation image will be compared to the preimage. The figure below shows that the image \begin{align*}A^\prime\end{align*} is a dilation by a scale factor of 2.

Dilations also need a center point. The center point is the center of the dilation. You use the center point to measure the distances to the preimage and the dilation image. It is these distances that determine the scale factor.

#### Let's describe the dilation in the diagram below:

The center of dilation is point \begin{align*}H\end{align*}.

Compare the lengths of corresponding sides to determine the scale factor. \begin{align*}\overline{IJ}\end{align*} is 2 units long and \begin{align*}\overline{I^\prime J^\prime}\end{align*} is 6 units long. \begin{align*}\frac{6}{2}=3\end{align*}, so the scale factor is 3. Therefore, the center point H is used to dilate \begin{align*}\triangle IJK\end{align*} to \begin{align*}\triangle I^\prime J^\prime K^\prime\end{align*} by a factor of 3.

#### Now, for the following measurements and the scale factors, let's determine the measure of the dilated images or preimages:

- \begin{align*}m \overline{A B} &= 15 \ cm \\ r &=\frac{1}{3}\end{align*}

Since we are given \begin{align*}m \overline{A B}\end{align*} , we need to find the measurements of the image \begin{align*}A^\prime B^\prime\end{align*}. You need to multiply the scale factor by the measure of \begin{align*}AB\end{align*} in order to find the measurement of the dilated image .

\begin{align*}m \overline{A^\prime B^\prime} = (r) m \overline{A B}\end{align*}

\begin{align*}& m \overline{A^\prime B^\prime}=\frac{1}{3}(15) \\ & m \overline{A^\prime B^\prime}= 5 \ cm \end{align*}

- \begin{align*}m \overline{H^\prime I^\prime} &= 24 \ cm \\ r &=2\end{align*}

Since you are given \begin{align*}m \overline{H^\prime I^\prime}\end{align*}, you need to find the measurements of the preimage \begin{align*}HI\end{align*}. Thus, you need to divide the scale factor by the measurement of \begin{align*}H^\prime I^\prime\end{align*} in order to find the measurement of the preimage.

\begin{align*}m \overline{H^\prime I^\prime} = (r) m \overline{H I}\end{align*}

\begin{align*}&24 = 2m\overline{H I} \\ & m \overline{H I}=\frac{24}{2} \\ & m \overline{H I}= 12 \ cm \end{align*}

### Examples

#### Example 1

Earlier, you were asked which one of the following figures represents a dilation and why:

You know that a dilation is a transformation that produces an image of the same shape but larger or smaller. Both of the figures above represent objects that involve dilations. In the figure with the triangles, the scale factor is 3.

The second figure with the squares also represents a dilation. In this figure, the center point \begin{align*}(3,-2)\end{align*} is used to dilate the small square by a factor of \begin{align*}2\end{align*}.

#### Example 2

Using the measurement below and the scale factor, determine the measure of the preimage.

\begin{align*}m \overline{T^\prime U^\prime}&=12 \ cm \\ r&= 4 \ cm \end{align*}

Here, you need to divide the scale factor by the measurement of \begin{align*}H^\prime I^\prime\end{align*} in order to find the measurement of the preimage \begin{align*}HI\end{align*}.

\begin{align*}m \overline{T^\prime U^\prime} = |r| m \overline{T U}\end{align*}

\begin{align*}&12 = 4m\overline{T U} \\ & m \overline{T U}=\frac{12}{4} \\ & m \overline{T U}= 3 \ cm \end{align*}

#### Example 3

Describe the dilation in the diagram below.

Look at the diagram below:

In the figure, the center point \begin{align*}D\end{align*} is used to dilate the \begin{align*}A\end{align*} by a factor of \begin{align*}\frac{1}{2}\end{align*}.

#### Example 4

Quadrilateral \begin{align*}STUV\end{align*} has vertices \begin{align*}S(-1, 3), T(2, 0), U(-2, -1),\end{align*} and \begin{align*}V(-3, 1)\end{align*}. The quadrilateral undergoes a dilation about the origin with a scale factor of \begin{align*}\frac{8}{5}\end{align*}. Sketch the preimage and the dilation image.

Look at the diagram below:

### Review

Find the measure of the dilation image given the following information:

- \begin{align*}m \overline{A B} &= 12 \ cm \\ r&=2\end{align*}

- \begin{align*}m \overline{C D} &= 25 \ cm \\ r&=\frac{1}{5}\end{align*}

- \begin{align*}m \overline{E F} &= 18 \ cm \\ r&=\frac{2}{3}\end{align*}

- \begin{align*}m \overline{G H} &= 18 \ cm \\ r&=3\end{align*}

- \begin{align*}m \overline{I J} &= 100 \ cm \\ r&=\frac{1}{10}\end{align*}

Find the measure of the preimage given the following information:

- \begin{align*}m \overline{K^\prime L^\prime} &= 48 \ cm \\ r&=4\end{align*}

- \begin{align*}m \overline{M^\prime N^\prime} &= 32 \ cm \\ r&=4\end{align*}

- \begin{align*}m \overline{O^\prime P^\prime} &= 36 \ cm \\ r&=6\end{align*}

- \begin{align*}m \overline{Q^\prime R^\prime} &= 20 \ cm \\ r&=\frac{1}{4}\end{align*}

- \begin{align*}m \overline{S^\prime T^\prime} &= 40 \ cm \\ r&=\frac{4}{5}\end{align*}

Describe the following dilations:

### Review (Answers)

To see the Review answers, open this PDF file and look for section 10.10.