What if you enlarged or reduced a triangle without changing its shape? How could you find the scale factor by which the triangle was stretched or shrunk? After completing this Concept, you'll be able to use the corresponding sides of dilated figures to solve problems like this one.
Watch This
CK-12 Foundation: Chapter7DilationA
Learn more about dilations by watching the video at this link.
Guidance
A transformation is an operation that moves, flips, or changes a figure to create a new figure. Transformations that preserve size are rigid and ones that do not are non-rigid. 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 \begin{align*}k\end{align*} and is always greater than zero. Also, if the original figure is labeled \begin{align*}\triangle ABC\end{align*}, for example, the dilation would be \begin{align*}\triangle A'B'C'\end{align*}. The ‘ indicates that it is a copy. This tic mark is said “prime,” so \begin{align*}A'\end{align*} is read “A prime.” A second dilation would be \begin{align*}A''\end{align*}, read “A double-prime.”
If the dilated image is smaller than the original, then the scale factor is \begin{align*}0<k<1\end{align*}.
If the dilated image is larger than the original, then the scale factor is \begin{align*}k>1\end{align*}.
Example A
The center of dilation is \begin{align*}P\end{align*} and the scale factor is 3. Find \begin{align*}Q'\end{align*}.
If the scale factor is 3 and \begin{align*}Q\end{align*} is 6 units away from \begin{align*}P\end{align*}, then \begin{align*}Q'\end{align*} is going to be \begin{align*}6 \times 3 = 18\end{align*} units away from \begin{align*}P\end{align*}. Because we are only dilating apoint, the dilation will be collinear with the original and center.
Example B
Using the picture above, change the scale factor to \begin{align*}\frac{1}{3}\end{align*}. Find \begin{align*}Q''\end{align*}.
Now the scale factor is \begin{align*}\frac{1}{3}\end{align*}, so \begin{align*}Q''\end{align*} is going to be \begin{align*}\frac{1}{3}\end{align*} the distance away from \begin{align*}P\end{align*} as \begin{align*}Q\end{align*} is. In other words, \begin{align*}Q''\end{align*} is going to be \begin{align*}6 \times \frac{1}{3} = 2\end{align*} units away from \begin{align*}P\end{align*}. \begin{align*}Q''\end{align*} will also be collinear with \begin{align*}Q\end{align*} and center.
Example C
\begin{align*}KLMN\end{align*} is a rectangle with length 12 and width 8. If the center of dilation is \begin{align*}K\end{align*} with a scale factor of 2, draw \begin{align*}K'L'M'N'\end{align*}.
If \begin{align*}K\end{align*} is the center of dilation, then \begin{align*}K\end{align*} and \begin{align*}K'\end{align*} will be the same point. From there, \begin{align*}L'\end{align*} will be 8 units above \begin{align*}L\end{align*} and \begin{align*}N'\end{align*} will be 12 units to the right of \begin{align*}N\end{align*}.
Watch this video for help with the Examples above.
CK-12 Foundation: Chapter7DilationB
Vocabulary
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 a 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
1. Find the perimeters of \begin{align*}KLMN\end{align*} and \begin{align*}K'L'M'N'\end{align*}. Compare this ratio to the scale factor.
2. \begin{align*}\triangle ABC\end{align*} is a dilation of \begin{align*}\triangle DEF\end{align*}. If \begin{align*}P\end{align*} is the center of dilation, what is the scale factor?
3. Find the scale factor, given the corresponding sides. In the diagram, the black figure is the original and \begin{align*}P\end{align*} is the center of dilation.
Answers:
1. The perimeter of \begin{align*}KLMN = 12 + 8 + 12 + 8 = 40\end{align*}. The perimeter of \begin{align*}K'L'M'N' = 24 + 16 + 24 + 16 = 80\end{align*}. The ratio is 80:40, which reduces to 2:1, which is the same as the scale factor.
2. Because \begin{align*}\triangle ABC\end{align*} is a dilation of \begin{align*}\triangle DEF\end{align*}, then \begin{align*}\triangle ABC \sim \triangle DEF\end{align*}. The scale factor is the ratio of the sides. Since \begin{align*}\triangle ABC\end{align*} is smaller than the original, \begin{align*}\triangle DEF\end{align*}, the scale factor is going to be less than one, \begin{align*}\frac{12}{20} = \frac{3}{5}\end{align*}.
If \begin{align*}\triangle DEF\end{align*} was the dilated image, the scale factor would have been \begin{align*}\frac{5}{3}\end{align*}.
3. Since the dilation is smaller than the original, the scale factor is going to be less than one. \begin{align*}\frac{8}{20}=\frac{2}{5}\end{align*}
Practice
In the two questions below, you are told the scale factor. Determine the dimensions of the dilation. In each diagram, the black figure is the original and \begin{align*}P\end{align*} is the center of dilation.
- \begin{align*}k = 4\end{align*}
- \begin{align*}k = \frac{1}{3}\end{align*}
In the question below, find the scale factor, given the corresponding sides. In the diagram, the black figure is the original and \begin{align*}P\end{align*} is the center of dilation.
- Find the perimeter of both triangles in #1. What is the ratio of the perimeters?
- Writing What happens if \begin{align*}k = 1\end{align*}?
Construction We can use a compass and straight edge to construct a dilation as well. Copy the diagram below.
- Set your compass to be \begin{align*}CG\end{align*} and use this setting to mark off a point 3 times as far from \begin{align*}C\end{align*} as \begin{align*}G\end{align*} is. Label this point \begin{align*}G'\end{align*}. Repeat this process for \begin{align*}CO\end{align*} and \begin{align*}CD\end{align*} to find \begin{align*}O'\end{align*} and \begin{align*}D'\end{align*}.
- Connect \begin{align*}G', O'\end{align*} and \begin{align*}D'\end{align*} to make \begin{align*}\triangle D'O'G'\end{align*}. Find the ratios, \begin{align*}\frac{D'O'}{DO}, \frac{O'G'}{OG}\end{align*} and \begin{align*}\frac{G'D'}{GD}\end{align*}.
- What is the scale factor of this dilation?
- Describe how you would dilate the figure by a scale factor of 4.
- Describe how you would dilate the figure by a scale factor of \begin{align*}\frac{1}{2}\end{align*}.
- The scale factor between two shapes is 1.5. What is the ratio of their perimeters?
- The scale factor between two shapes is 1.5. What is the ratio of their areas? Hint: Draw an example and calculate what happens.
- Suppose you dilate a triangle with side lengths 3, 7, and 9 by a scale factor of 3. What are the side lengths of the image?
- Suppose you dilate a rectangle with a width of 10 and a length of 12 by a scale factor of \begin{align*}\frac{1}{2}\end{align*}. What are the dimensions of the image?
- Find the areas of the rectangles in #14. What is the ratio of their areas?