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Dilation

Larger or smaller version of a figure that preserves its shape.

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Dilation

Dilation

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 k\begin{align*}k\end{align*} and is always greater than zero. Also, if the original figure is labeled ABC\begin{align*}\triangle ABC\end{align*}, for example, the dilation would be ABC\begin{align*}\triangle A'B'C'\end{align*}. The ‘ indicates that it is a copy. This tic mark is said “prime,” so A\begin{align*}A'\end{align*} is read “A prime.” A second dilation would be A′′\begin{align*}A''\end{align*}, read “A double-prime.”

If the dilated image is smaller than the original, then the scale factor is 0<k<1\begin{align*}0 < k < 1\end{align*}.

If the dilated image is larger than the original, then the scale factor is k>1\begin{align*}k > 1\end{align*}.

Dilating a Point

1. The center of dilation is P\begin{align*}P\end{align*} and the scale factor is 3. Find Q\begin{align*}Q'\end{align*}.

If the scale factor is 3 and Q\begin{align*}Q\end{align*} is 6 units away from P\begin{align*}P\end{align*}, then Q\begin{align*}Q'\end{align*} is going to be 6×3=18\begin{align*}6 \times 3 = 18\end{align*} units away from P\begin{align*}P\end{align*}. Because we are only dilating a point, the dilation will be collinear with the original and center.

2. Using the picture above, change the scale factor to 13\begin{align*}\frac{1}{3}\end{align*}. Find Q′′\begin{align*}Q''\end{align*}.

Now the scale factor is 13\begin{align*}\frac{1}{3}\end{align*}, so Q′′\begin{align*}Q''\end{align*} is going to be 13\begin{align*}\frac{1}{3}\end{align*} the distance away from P\begin{align*}P\end{align*} as Q\begin{align*}Q\end{align*} is. In other words, Q′′\begin{align*}Q''\end{align*} is going to be 6×13=2\begin{align*}6 \times \frac{1}{3} = 2\end{align*} units away from P\begin{align*}P\end{align*}. Q′′\begin{align*}Q''\end{align*} will also be collinear with Q\begin{align*}Q\end{align*} and center.

Drawing a Dilation

KLMN\begin{align*}KLMN\end{align*} is a rectangle with length 12 and width 8. If the center of dilation is K\begin{align*}K\end{align*} with a scale factor of 2, draw KLMN\begin{align*}K'L'M'N'\end{align*}.

If K\begin{align*}K\end{align*} is the center of dilation, then K\begin{align*}K\end{align*} and K\begin{align*}K'\end{align*} will be the same point. From there, L\begin{align*}L'\end{align*} will be 8 units above L\begin{align*}L\end{align*} and N\begin{align*}N'\end{align*} will be 12 units to the right of N\begin{align*}N\end{align*}.

Examples

Example 1

Find the perimeters of KLMN\begin{align*}KLMN\end{align*} and KLMN\begin{align*}K'L'M'N'\end{align*}. Compare this ratio to the scale factor.

The perimeter of KLMN=12+8+12+8=40\begin{align*}KLMN = 12 + 8 + 12 + 8 = 40\end{align*}. The perimeter of KLMN=24+16+24+16=80\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.

Example 2

ABC\begin{align*}\triangle ABC\end{align*} is a dilation of DEF\begin{align*}\triangle DEF\end{align*}. If P\begin{align*}P\end{align*} is the center of dilation, what is the scale factor?

Because ABC\begin{align*}\triangle ABC\end{align*} is a dilation of DEF\begin{align*}\triangle DEF\end{align*}, then ABCDEF\begin{align*}\triangle ABC \sim \triangle DEF\end{align*}. The scale factor is the ratio of the sides. Since ABC\begin{align*}\triangle ABC\end{align*} is smaller than the original, DEF\begin{align*}\triangle DEF\end{align*}, the scale factor is going to be less than one, 1220=35\begin{align*}\frac{12}{20} = \frac{3}{5}\end{align*}

If DEF\begin{align*}\triangle DEF\end{align*} was the dilated image, the scale factor would have been \begin{align*}\frac{5}{3}\end{align*}.

Example 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.

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*}

Review

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.

1. \begin{align*}k = 4\end{align*}
2. \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.

1. Find the perimeter of both triangles in #1. What is the ratio of the perimeters?
2. 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.

1. 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*}.
2. 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*}.
3. What is the scale factor of this dilation?
4. Describe how you would dilate the figure by a scale factor of 4.
5. Describe how you would dilate the figure by a scale factor of \begin{align*}\frac{1}{2}\end{align*}.
1. The scale factor between two shapes is 1.5. What is the ratio of their perimeters?
2. The scale factor between two shapes is 1.5. What is the ratio of their areas? Hint: Draw an example and calculate what happens.
3. 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?
4. 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?
5. Find the areas of the rectangles in #14. What is the ratio of their areas?

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Color Highlighted Text Notes

Vocabulary Language: English

Dilation

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

A quadrilateral is a closed figure with four sides and four vertices.

Ratio

A ratio is a comparison of two quantities that can be written in fraction form, with a colon or with the word “to”.

Scale Factor

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

Transformation

A transformation moves a figure in some way on the coordinate plane.

Vertex

A vertex is a point of intersection of the lines or rays that form an angle.

Rigid Transformation

A rigid transformation is a transformation that preserves distance and angles, it does not change the size or shape of the figure.

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