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Grades 9-12

9.01 Dilations

Lesson
Practice

Introduction

In 7th grade, we saw scale drawings being produced at different scales. In 8th grade, we applied scale factor to figures to enlarge or reduce their size. We will practice using scale factor to dilate figures here and extend our understanding of how the center of dilation effects the image of a figure.

Dilations

Recall that a dilation is a transformation which changes the size of a figure through either an enlargement, which makes the figure bigger, or a reduction, which makes the figure smaller, by a given scale factor.

Dilation

A proportional increase or decrease in size in all directions.

There are two quadrilaterals of the same shape but different sizes. The smaller quadrilateral is in the larger quadrilateral. Dashed arrows connect the corresponding vertices.
Scale factor

The constant, k, that is multiplied by the length of each side of a figure to produce an image whose segments are k times the size of the corresponding segments in the pre-image.

When performed on the coordinate plane, a dilation will have a specified scale factor as well as a specified center of dilation. If none is specified the origin is assumed to be the center of dilation.

Center of dilation

A fixed point on the coordinate plane about which a figure is either enlarged or reduced.

Consider the following dilation, where pre-image A is dilated by a factor of k around point P.

A figure showing pre image A, triangle X Y Z being dilated by a factor of k around a point P. The resulting image is X prime Y prime Z prime. Speak to your teacher for more details.
  • The distance between the center of a dilation and any point on the image is equal to the scale factor multiplied by the distance between the dilation center and the corresponding point on the pre-image: For example, PX'=k \cdot PX
  • The length of the image of a line segment is equal to the length of the line segment multiplied by the scale factor: For example, X'Y'=k \cdot XY
  • Notice that point P is not on any of the line segments of the pre-image. In this case, the pre-image and the image will be parallel
  • Since each corresponding segment on the pre-image and the image will be parallel, the angles formed where the segments meet at each vertex will have the same angle measure. Thus, corresponding angles in the image and pre-image are congruent.

Coordinate form: The dilation \left(x,y\right) \to \left(kx,ky\right) takes the pre-image and dilates it by a factor of k, about the origin.

Function notation: The dilation D_{k,P}(A) takes the pre-image, A, and dilates it by a factor of k, with a center of dilation P.

If k>1, the dilation will be an enlargment, and if 0<k<1, the dilation will be a reduction. If k=1, the dilation maps the pre-image onto itself.

Examples

Example 1

Consider the figure shown on the coordinate grid:

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Dilate the figure using the rule (x,y) \to (4x, 4y).

Worked Solution
Create a strategy

Since the center of dilation is the origin, (0,0), and the scale factor is 4 based on the coordinate mapping, we can take the coordinates of each point in the pre-image and multiply them by the scale factor to get the vertices of the image.

Apply the idea

We have:

(4, 8) \to (16, 32)

(10, 8) \to (40, 32)

(4, 14) \to (16, 56)

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Describe how the pre-image and its image are related.

Worked Solution
Create a strategy

Using the figure and its image, connect the corresponding vertices of the pre-image and image. Also, compare corresponding segments between pre-image and image.

Apply the idea
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We would expect that the image would be 4 times farther from the center of dilation than the pre-image is, which appears to be true based on the graph.

We should also expect the segment lengths of A'B'C' to be 4 times longer than the corresponding segments of ABC. We can spot check one of the segments to verify this:AB = 10 - 4 = 6 \text{ and } A'B' = 40-16 = 24 \text{ and } 6 \times 4 = 24

We can see that corresponding segements are parallel to one another: \overline{AB} is a horizontal line with a slope of 0, as is \overline{A'B'}. We can see both \overline{AC} and \overline{A'C'} are vertical with no slope. Both \overline{CB} and \overline{C'B'} are decreasing lines with a slope of -1.

Example 2

Find the scale factor for the following dilation:

Two parallelograms are drawn. Parallelogram A B C D has side B C of length 16 and side D C of length 32. Segment A B and segment D C are marked parallel as well as segment B C and segment A D. Parallelogram A prime B prime C prime and D prime has side B prime C prime of length 2 and segment D prime C prime of length 4.
Worked Solution
Create a strategy

We can see that the pre-image has side lengths of 32 and 16, and the image has side lengths of 4 and 2. This indicates that the pre-image has been reduced. To find the scale factor we can divide one of the lengths of the image by the corresponding side length of the pre-image.

Apply the idea

\dfrac{C'D'}{CD}=\dfrac{4}{32} Simplifying the quotient gives a scale factor of \dfrac{1}{8}.

Example 3

Consider the figure shown on the coordinate grid:

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Dilate the figure using the function rule D_{4, (7,8)}(\triangle ABC).

Worked Solution
Create a strategy

The function notation means that the center of dilation is located at \left(7, 8 \right) and the scale factor is four. Determine translations from the figure's vertices to the center of dilation, apply the scale factor to each translation, and perform the new translations to the center of dilation to plot the image.

Apply the idea

We can determine the translation from each vertex to the center of dilation, then apply the scale factor to the translations.

  1. Determine the translations from the center of dilation to the vertices of the figure:
    • \left(7,8\right)\to\left(4,8\right): translate 3 units left and 0 units up or down
    • \left(7,8\right)\to\left(10,8\right): translate 3 units right and 0 units up or down
    • \left(7,8\right)\to\left(4,14\right): translate 3 units left and 6 units up
  2. Multiply these translations by the scale factor, 4, to find the images of these points under the dilation in relation to the center of dilation:
    • Translate 12 units left and 0 units up or down
    • Translate 12 units right and 0 units up or down
    • Translate 12 units left and 24 units up

From the center of dilation, (7,8), we can plot the vertices of the image after multiplying the translation from each vertex to the center of dilation by the scale factor:

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Compare the effects of the dilation when the center of dilation is on a line segment of the pre-image to when the center of dilation is not on the pre-image.

Worked Solution
Create a strategy

Consider the location of the center of dilation when the center of dilation is a point that does not lie on the pre-image and when the center of dilation is a point that lies on the pre-image.

Apply the idea

Since the center of dilation lies on \overline{AB}, it also lines on the corresponding \overline{A'B'}.

We can see how the pre-image was enlarged by drawing segments from the center of dilation to each vertex on the pre-image and its image.

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When the center of dilation is not a point on the pre-image, as shown in worked question 1, the line segments of the image are parallel to the line segments of the pre-image. In this case, when the center of dilation lies on one of the pre-image segments, \overline{AB}, we see that the corresponding segment \overline{A'B'} is also on the same line.

Example 4

Consider the figure on the coordinate plane shown:

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Apply the dilation D_{\frac{1}{2},\left(6,7\right)} to the figure.

Worked Solution
Create a strategy

We can determine the translation from each vertex to the center of dilation, then apply the scale factor to the translations.

Apply the idea
  1. Determine the translations from the center of dilation to the vertices of the figure:
    • \left(6,7\right)\to\left(2,1\right): translate 4 units left and 6 units down
    • \left(6,7\right)\to\left(8,3\right): translate 2 units right and 4 units down
    • \left(6,7\right)\to\left(12,9\right): translate 6 units right and 2 units up
    • \left(6,7\right)\to\left(4,11\right): translate 2 units left and 4 units up
  2. Multiply these translations by the scale factor, \dfrac{1}{2}, to find the images of these points under the dilation in relation to the center of dilation:
    • Translate 2 units left and 3 units down
    • Translate 1 unit right and 2 units down
    • Translate 3 units right and 1 unit up
    • Translate 1 unit left and 2 units up
  3. Apply these translations to the center of dilation, \left(6,7\right), to determine the coordinates of the images of the vertices under dilation:
    • Translating \left(6,7\right) by 2 units left and 3 units down gives us \left(4,4\right)
    • Translating \left(6,7\right) by 1 unit right and 2 units down gives us \left(7,5\right)
    • Translating \left(6,7\right) by 3 units right and 1 unit up gives us \left(9,8\right)
    • Translating \left(6,7\right) by 1 unit left and 2 units up gives us \left(5,9\right)
  4. Plot the image of the dilation, using the images of the vertices determined in the previous step.
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Compare the pre-image with its image after performing the dilation.

Worked Solution
Create a strategy

Use the figure and its image sketched in part (a) and what we know about dilating a figure around a central point.

Apply the idea

The pre-image and its image have the same orientation. The pre-image and its image are not congruent because the side lengths of the image are not preserved with a dilation.

The side lengths of the image are parallel to the side lengths of the pre-image.

Since all corresponding side lengths are parallel, each pair of corresponding interior angles will be congruent.

The scale factor is \dfrac{1}{2}, meaning that the side lengths of the image should be half the length of the pre-image side lengths. This visually appears to be true.

The distance between the center of a dilation and any point on the image appears to be equal to half of the distance between the dilation center and the corresponding point on the pre-image, shown on the grid below.

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Reflect and check

We could calculate slopes and use the distance formula to verify each segment matches the given criteria. Or, we could use a ruler and protractor to verify corresponding segments are an equal distance apart (thus parallel), that the length of each segment on the image is \dfrac{1}{2} the size of the corresponding segemnt of the pre-image, and that corresponding angles are congruent.

Idea summary

The dilation D_{k,P}(A) takes the pre-image A and dilates it by a factor of k with a center of dilation at point P. When this dilation occurs:

  • The distance between P and any point on the image will be k times the size of the distance between P and the corresponding point on A
  • The length of the image segments will be k times the size of the corresponding segment lengths of A
  • When P is not a point on A, the image will be parallel to A
  • When P lies on a line segment of A, its image will lie on the same line

Outcomes

G.SRT.A.1

Verify experimentally the properties of dilations given by a center and a scale factor.

G.SRT.A.1.A

A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

G.SRT.A.1.B

The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

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