Quadrilateral \begin{align*}WXYZ\end{align*} has coordinates \begin{align*}W(-5, -5), X(-2, 0), Y(2, 3)\end{align*} and \begin{align*}Z(-1, 3)\end{align*}. Draw the quadrilateral on the Cartesian plane.
The quadrilateral undergoes a dilation centered at the origin of scale factor \begin{align*}\frac{1}{3}\end{align*}. Show the resulting image.
Watch This
First watch this video to learn about graphs of dilations.
CK-12 Foundation Chapter10GraphsofDilationsA
Then watch this video to see some examples.
CK-12 Foundation Chapter10GraphsofDilationsB
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.
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.
Example A
Line \begin{align*}\overline{A B}\end{align*} drawn from (-4, 2) to (3, 2) has undergone a dilation about the origin to produce \begin{align*}A^\prime(-6, 3)\end{align*} and \begin{align*}B^\prime(4.5, 3)\end{align*}. Draw the preimage and dilation image and determine the scale factor.
Solution:
\begin{align*}& scale \ factor = \frac{dilation \ image \ length}{preimage \ length} \\ & scale \ factor = \frac{10.5}{7.0} \\ & scale \ factor = \frac{3}{2}\end{align*}
Example B
The diamond \begin{align*}ABCD\end{align*} undergoes a dilation about the origin to form the image \begin{align*}A^\prime B^\prime C^\prime D^\prime\end{align*}. Find the coordinates of the dilation image. Using the diagram, determine the scale factor.
Solution:
\begin{align*}& scale \ factor = \frac{dilation \ image \ length}{preimage \ length}\\ & scale factor = \frac{7.21}{3.61} \\ & scale factor = 2\end{align*}
Example C
The diagram below undergoes a dilation about the origin to form the dilation image. Find the coordinates of \begin{align*}A\end{align*} and \begin{align*}B\end{align*} and \begin{align*}A^\prime\end{align*} and \begin{align*}B^\prime\end{align*} of the dilation image. Using the diagram, determine the scale factor.
Solution:
\begin{align*}& scale factor = \frac{dilation \ image \ length}{preimage \ length}\\ & scale \ factor = \frac{2.00}{10.00} \\ & scale \ factor = \frac{1}{5}\end{align*}
Concept Problem Revisited
Test to see if the dilation is correct by determining the scale factor.
\begin{align*}& scale \ factor = \frac{dilation \ image \ length}{preimage \ length} \\ & scale \ factor = \frac{10.63}{3.54} \\ & scale \ factor = 3\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. Line \begin{align*}\overline{S T}\end{align*} 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*}. Draw the preimage and image and properly label each.
2. The polygon below has undergone a dilation about the origin with a scale factor of \begin{align*}\frac{5}{3}\end{align*}. Draw the dilation image and properly label each.
3. The triangle with vertices \begin{align*}J(-5, -2), K(-1, 4)\end{align*} and \begin{align*}L(1, -3)\end{align*} has undergone a dilation of scale factor \begin{align*}\frac{1}{2}\end{align*}. about the center point \begin{align*}L\end{align*}. Draw and label the dilation image and the preimage then check the scale factor.
Answers:
1.
\begin{align*}& scale \ factor = \frac{dilation \ image \ length}{preimage \ length} \\ & scale \ factor = \frac{12.00}{4.00} \\ & scale \ factor = 3\end{align*}
2.
\begin{align*}& scale factor = \frac{dilation \ image \ length}{preimage \ length} \\ & scale \ factor = \frac{5.00}{3.00} \\ & scale \ factor = \frac{5}{3}\end{align*}
3.
\begin{align*}& scale \ factor = \frac{dilation \ image \ length}{preimage \ length}\\ & scale \ factor = \frac{7.21}{3.61} \\ & scale \ factor = \frac{1}{2}\end{align*}
Practice
- Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about the origin.
- Dilate the above figure by a factor of \begin{align*}2\end{align*} about point D.
- Dilate the above figure by a factor of \begin{align*}3\end{align*} about the origin.
- Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about point C.
- Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about the origin.
- Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about point C.
- Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about the origin.
- Dilate the above figure by a factor of \begin{align*}\frac{1}{4}\end{align*} about point C.
- Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about the origin.
- Dilate the above figure by a factor of \begin{align*}2\end{align*} about point A.
- Dilate the above figure by a factor of \begin{align*}2\end{align*} about the origin.
- Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about point D.
- Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about the origin.
- Dilate the above figure by a factor of \begin{align*}3\end{align*} about point D.
- Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about the origin.
- Dilate the above figure by a factor of \begin{align*}\frac{1}{2}\end{align*} about point C.