Which one of the following figures represents a dilation? Explain.
Watch This
First watch this video to learn about dilations.
CK12 Foundation Chapter10DilationsA
Then watch this video to see some examples.
CK12 Foundation Chapter10DilationsB
Guidance
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.
Example A
Describe the dilation in the diagram below. The center of dilation is point \begin{align*}H\end{align*}.
Solution:
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.
Example B
Using the measurement below and the scale factor, determine the measure of the dilated image.
\begin{align*}m \overline{A B} &= 15 \ cm \\ r &=\frac{1}{3}\end{align*}
Solution: 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*}A^\prime B^\prime\end{align*}.
\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*}
Example C
Using the measurement below and the scale factor, determine the measure of the preimage.
\begin{align*}m \overline{H^\prime I^\prime} &= 24 \ cm \\ r &=3\end{align*}
Solution: 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{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*}
Concept Problem Revisited
Which one of the following figures represents a dilation? Explain.
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*}.
Vocabulary
 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.
 Dilation
 A dilation is a transformation that enlarges or reduces the size of a figure.
 Scale Factor
 The scale factor determines how much bigger or smaller the dilation image will be compared to the preimage. The scale factor often uses the symbol \begin{align*}r\end{align*}.
 Image
 In a transformation, the final figure is called the image.
 Preimage
 In a transformation, the original figure is called the preimage.
 Transformation
 A transformation is an operation that is performed on a shape that moves or changes it in some way. There are four types of transformations: translations, reflections, dilations and rotations.
Guided Practice
1. 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*}
2. Describe the dilation in the diagram below.
3. 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.
Answers:
1. 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 = 2m\overline{T U} \\ & m \overline{T U}=\frac{12}{4} \\ & m \overline{T U}= 3 \ cm \end{align*}
2. 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*}.
3. Look at the diagram below:
Practice
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: