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Dilation in the Coordinate Plane

Multiplication of coordinates by a scale factor.

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Graphs of Dilations

Quadrilateral \begin{align*}WXYZ\end{align*}WXYZ has coordinates \begin{align*}W \ (\text{-}5, \text{-}5), \ X \ (\text{-}2, 0), \ Y \ (2, 3),\end{align*}W (-5,-5), X (-2,0), Y (2,3), and \begin{align*}Z \ (\text{-}1, 3).\end{align*}Z (-1,3). Draw the quadrilateral on the Cartesian plane.

Suppose the quadrilateral undergoes a dilation centered at the origin of scale factor \begin{align*}\frac{1}{3}.\end{align*}13. What is  the resulting image?

Graphs of 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 pre-image) to create a new figure (called the image). The scale factor, r, determines how much bigger or smaller the dilated image will be compared to the preimage.

In order to graph a dilation, use the center of dilation and the scale factor. Find the distance between a point on the preimage and the center of dilation. Multiply this length by the scale factor. The corresponding point on the image will be this distance away from the center of dilation in the same direction as the original point.

If you compare the length of a side on the preimage to the length of the corresponding side on the image, the length of the side on the image will be the length of the side on the preimage multiplied by the scale factor.

 

 

Draw the preimage and image for the following dilations and determine the scale factor:

Line \begin{align*}\overline{A B}\end{align*}AB¯¯¯¯¯¯¯¯ drawn from (-4, 2) to (3, 2) has undergone a dilation about the origin to produce \begin{align*}A^\prime(-6, 3)\end{align*}A(6,3) and \begin{align*}B^\prime(4.5, 3)\end{align*}B(4.5,3)

\begin{align*}& \mathrm{scale \ factor} = \frac{\mathrm{dilation \ image \ length}}{\mathrm{preimage \ length}} \\ & \mathrm{scale \ factor} = \frac{10.5}{7.0} \\ & \mathrm{scale \ factor} = \frac{3}{2}\end{align*}scale factor=dilation image lengthpreimage lengthscale factor=10.57.0scale factor=32

Determine the coordinates and the scale factor for the following dilations:

  1. The diamond \begin{align*}ABCD\end{align*}ABCD undergoes a dilation about the origin to form the image \begin{align*}A^\prime B^\prime C^\prime D^\prime\end{align*}ABCD

\begin{align*}& \mathrm{scale \ factor} = \frac{\mathrm{dilation \ image \ length}}{\mathrm{preimage \ length}}\\ & \mathrm{scale factor} = \frac{7.21}{3.61} \\ & \mathrm{scale factor} = 2\end{align*}scale factor=dilation image lengthpreimage lengthscalefactor=7.213.61scalefactor=2

  1. The diagram below undergoes a dilation about the origin to form the dilation image. 

\begin{align*}& \mathrm{scale factor }= \frac{\mathrm{dilation \ image \ length}}{\mathrm{preimage \ length}}\\ & \mathrm{scale \ factor }= \frac{2.00}{10.00} \\ & \mathrm{scale \ factor }= \frac{1}{5}\end{align*}scalefactor=dilation image lengthpreimage lengthscale factor=2.0010.00scale factor=15

Examples

Example 1

Earlier, you were told that quadrilateral \begin{align*}WXYZ\end{align*}WXYZ has coordinates \begin{align*}W \ (\text{-}5, \text{-}5), \ X \ (\text{-}2, 0), \ Y\ (2, 3),\end{align*}W (-5,-5), X (-2,0), Y (2,3), and \begin{align*}Z \ (\text{-}1, 3).\end{align*}Z (-1,3). If the quadrilateral undergoes a dilation centered at the origin of scale factor \begin{align*}\frac{1}{3},\end{align*}13,  what is the resulting image?

Test to see if the dilation is correct by determining the scale factor.

\begin{align*}& \mathrm{scale \ factor} = \frac{\mathrm{dilation \ image \ length}}{\mathrm{preimage \ length}} \\ & \mathrm{scale \ factor} = \frac{10.63}{3.54} \\ & \mathrm{scale \ factor} = 3\end{align*}scale factor=dilation image lengthpreimage lengthscale factor=10.633.54scale factor=3

Example 2

Line \begin{align*}\overline{S T}\end{align*}ST¯¯¯¯¯¯¯ drawn from (-3, 4) to (-3, 8) has undergone a dilation of scale factor 3 about the point \begin{align*}A \ (1, 6).\end{align*}A (1,6). Draw the preimage and image and properly label each.

\begin{align*}& \mathrm{scale \ factor} = \frac{\mathrm{dilation \ image \ length}}{\mathrm{preimage \ length}} \\ & \mathrm{scale \ factor} = \frac{12.00}{4.00} \\ & \mathrm{scale \ factor} = 3\end{align*}scale factor=dilation image lengthpreimage lengthscale factor=12.004.00scale factor=3

Example 3

The polygon below has undergone a dilation about the origin with a scale factor of \begin{align*}\frac{5}{3}.\end{align*}53. Draw the dilation image and properly label each point.

\begin{align*}& \mathrm{scale \ factor} = \frac{\mathrm{dilation \ image \ length}}{\mathrm{preimage \ length}} \\ & \mathrm{scale \ factor} = \frac{5.00}{3.00} \\ & \mathrm{scale \ factor} = \frac{5}{3}\end{align*}scale factor=dilation image lengthpreimage lengthscale factor=5.003.00scale factor=53

Example 4

The triangle with vertices \begin{align*}J \ (\text{-}5, -2), \ K \ (\text{-}1, 4),\end{align*}J (-5,2), K (-1,4), and \begin{align*}L \ (1, \text{-}3)\end{align*}L (1,-3) has undergone a dilation of scale factor \begin{align*}\frac{1}{2}\end{align*}12 about the center point \begin{align*}L.\end{align*}L. Draw and label the dilation image and the preimage, then verify the scale factor.

\begin{align*}& \mathrm{scale \ factor} = \frac{\mathrm{dilation \ image \ length}}{\mathrm{preimage \ length}}\\ & \mathrm{scale \ factor} = \frac{7.21}{3.61} \\ & \mathrm{scale \ factor} = \frac{1}{2}\end{align*}scale factor=dilation image lengthpreimage lengthscale factor=7.213.61scale factor=12

Review

  1. Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*}12 about the origin.
  2. Dilate the above figure by a factor of \begin{align*}2\end{align*}2 about point \begin{align*}D.\end{align*}D.

  1. Dilate the above figure by a factor of \begin{align*}3\end{align*}3 about the origin.
  2. Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about point \begin{align*}C.\end{align*} 

  1. Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about the origin.
  2. Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about point \begin{align*}C.\end{align*} 

  1. Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about the origin.
  2. Dilate the above figure by a factor of \begin{align*}\frac{1}{4}\end{align*} about point \begin{align*}C.\end{align*} 

  1. Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about the origin.
  2. Dilate the above figure by a factor of \begin{align*}2\end{align*} about point \begin{align*}A.\end{align*} 

  1. Dilate the above figure by a factor of \begin{align*}2\end{align*} about the origin.
  2. Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about point \begin{align*}D.\end{align*} 

  1. Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about the origin.
  2. Dilate the above figure by a factor of \begin{align*}3\end{align*} about point \begin{align*}D.\end{align*} 

  1. Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about the origin.
  2. Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about point \begin{align*}C.\end{align*} 

Review (Answers)

To see the Review answers, open this PDF file and look for section 10.11. 

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Vocabulary

Dilation

To reduce or enlarge a figure according to a scale factor is a dilation.

Distance Formula

The distance between two points (x_1, y_1) and (x_2, y_2) can be defined as d= \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}.

Mapping

Mapping is a procedure involving the plotting of points on a coordinate grid to see the behavior of a function.

Scale Factor

A scale factor is a ratio of the scale to the original or actual dimension written in simplest form.

Image Attributions

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