What if you were given the coordinates of a figure and were asked to dilate that figure by a scale factor of 2? How could you find the coordinates of the dilated figure? After completing this Concept, you'll be able to solve problems like this one.
Watch This
CK12 Foundation: Chapter7DilationintheCoordinatePlaneA
Guidance
A dilation makes a figure larger or smaller, but has the same shape as the original. In other words, the dilation is similar to the original. All dilations have a center and a scale factor. The center is the point of reference for the dilation (like the vanishing point in a perspective drawing) and scale factor tells us how much the figure stretches or shrinks. A scale factor is typically labeled
If the dilated image is smaller than the original, then the scale factor is
If the dilated image is larger than the original, then the scale factor is
To dilate something in the coordinate plane, multiply each coordinate by the scale factor. This is called mapping. For any dilation the mapping will be
Example A
Determine the coordinates of
The coordinates of
Again, the center, original point, and dilated point are collinear. Therefore, you can draw a ray from the origin to
Example B
Show that dilations preserve shape by using the distance formula. Find the lengths of the sides of both triangles in Example A.
From this, we also see that all the sides of
Example C
Quadrilateral
Remember that to dilate something in the coordinate plane, multiply each coordinate by the scale factor.
For this dilation, the mapping will be
Watch this video for help with the Examples above.
CK12 Foundation: Chapter7DilationintheCoordinatePlaneB
Vocabulary
In the graph above, the blue quadrilateral is the original and the red image is the dilation. A dilation an enlargement or reduction of a figure that preserves shape but not size. All dilations are similar to the original figure. Similar figures are the same shape but not necessarily the same size. The center of dilation is the point of reference for the dilation and the scale factor for a dilation tells us how much the figure stretches or shrinks.
Guided Practice
Given
1.
2.
3.
Answers
Remember that the mapping will be
1.
2.
3.
Practice
Given

A(8,2),A′(12,3) 
A(−5,−9),A′(−45,−81) 
A(22,−7),A′(11,−3.5)
The origin is the center of dilation. Find the coordinates of the dilation of each figure, given the scale factor.

A(2,4),B(−3,7),C(−1,−2);k=3 
A(12,8),B(−4,−16),C(0,10);k=34
MultiStep Problem Questions 612 build upon each other.
 Plot
A(1,2),B(12,4),C(10,10) . Connect to form a triangle.  Make the origin the center of dilation. Draw 4 rays from the origin to each point from #6. Then, plot
A′(2,4),B′(24,8),C′(20,20) . What is the scale factor?  Use
k=4 , to findA′′B′′C′′ . Plot these points.  What is the scale factor from
A′B′C′ to \begin{align*}A''B''C''\end{align*}?  Find (\begin{align*}O\end{align*} is the origin):
 \begin{align*}OA\end{align*}
 \begin{align*}AA'\end{align*}
 \begin{align*}AA''\end{align*}
 \begin{align*}OA'\end{align*}
 \begin{align*}OA''\end{align*}
 Find:
 \begin{align*}AB\end{align*}
 \begin{align*}A'B'\end{align*}
 \begin{align*}A''B''\end{align*}
 Compare the ratios:
 \begin{align*}OA:OA'\end{align*} and \begin{align*}AB:A'B'\end{align*}
 \begin{align*}OA:OA''\end{align*} and \begin{align*}AB:A''B''\end{align*}
For questions 1318, use quadrilateral \begin{align*}ABCD\end{align*} with \begin{align*}A(1, 5), B(2, 6), C(3, 3)\end{align*} and \begin{align*}D(1, 3)\end{align*} and its transformation \begin{align*}A'B'C'D'\end{align*} with \begin{align*}A'(3, 1), B'(0, 4), C'(3, 5)\end{align*} and \begin{align*}D'(3, 5)\end{align*}.
 Plot the two quadrilaterals in the coordinate plane.
 Find the equation of \begin{align*}\overleftrightarrow{CC'}\end{align*}.
 Find the equation of \begin{align*}\overleftrightarrow{DD'}\end{align*}.
 Find the intersection of these two lines algebraically or graphically.
 What is the significance of this point?
 What is the scale factor of the dilation?