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7.15: Dilation of a Shape

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

Dilation

Two figures are similar if they are the same shape but not necessarily the same size. One way to create similar figures is by dilating. A dilation makes a figure larger or smaller but the new resulting figure has the same shape as the original.

Dilation: An enlargement or reduction of a figure that preserves shape but not size. All dilations are similar to the original figure.

Dilations have a center and a scale factor. The center is the point of reference for the dilation and the scale factor tells us how much the figure stretches or shrinks. A scale factor is labeled k. Only positive scale factors, k, will be considered in this text.

If the dilated image is smaller than the original, then 0<k<1.

If the dilated image is larger than the original, then k>1.

A dilation, or image, is always followed by a .

Label It Say It
“prime” (copy of the original)
A “a prime” (copy of point A)
A “a double prime” (second copy)

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?

Example 7.15.1

Find the perimeters of KLMN and KLMN. Compare this ratio to the scale factor.

f-d_f0c1594c2bbb61c374ba5c3748f8b381cad830b33ed6e181f953b74f+IMAGE_TINY+IMAGE_TINY.png
Figure 7.15.1

Solution

The perimeter of KLMN=12+8+12+8=40. The perimeter of KLMN=24+16+24+16=80. The ratio is 80:40, which reduces to 2:1, which is the same as the scale factor.

Example 7.15.2

f-d_2709ad03596ee418429052b007d74e83ca7ac308d93121f5b8770ae1+IMAGE_TINY+IMAGE_TINY.png
Figure 7.15.2

ΔABC is a dilation of ΔDEF. If P is the center of dilation, what is the scale factor?

Solution

f-d_570c7109d64e5751cfd19be0d28f8594b0a9b819ef9a85c0f983daca+IMAGE_TINY+IMAGE_TINY.png
Figure 7.15.3

Because ΔABC is a dilation of ΔDEF, then ΔABCΔDEF. The scale factor is the ratio of the sides. Since ΔABC is smaller than the original, ΔDEF, the scale factor is going to be less than one, 1220=35.

If ΔDEF was the dilated image, the scale factor would have been 53.

Example 7.15.3

The center of dilation is P and the scale factor is 3.

Find Q.

f-d_394fbefb3f6f9cfff1510d015488fe115a16e0345bb09484533dfade+IMAGE_TINY+IMAGE_TINY.png
Figure 7.15.4

Solution

If the scale factor is 3 and Q is 6 units away from P, then Q is going to be 6×3=18 units away from P. The dilated image will be on the same line as the original image and center.

f-d_38b563dc51e584a7af5c32142b6a9ded9af7ab7ea9e3cd5dbdb2a693+IMAGE_TINY+IMAGE_TINY.png
Figure 7.15.5

Example 7.15.4

Using the picture above, change the scale factor to 13.

Find Q using this new scale factor.

Solution

The scale factor is 13, so Q is going to be 6×13=2 units away from P. Q will also be collinear with Q and center.

f-d_4e1487a277f7f9bc945c85f6a2bb46034f63b80376498f10c768a94b+IMAGE_TINY+IMAGE_TINY.png
Figure 7.15.6

Example 7.15.5

KLMN is a rectangle. If the center of dilation is K and k=2, draw KLMN.

f-d_f0c1594c2bbb61c374ba5c3748f8b381cad830b33ed6e181f953b74f+IMAGE_TINY+IMAGE_TINY.png
Figure 7.15.7

Solution

If K is the center of dilation, then K and K will be the same point. From there, L will be 8 units above L and N will be 12 units to the right of N.

f-d_2709ad03596ee418429052b007d74e83ca7ac308d93121f5b8770ae1+IMAGE_TINY+IMAGE_TINY.png
Figure 7.15.8

Review

For the given shapes, draw the dilation, given the scale factor and center.

  1. k=3.5, center is A
f-d_f2a8e26a77124b145795e9d848ad2478e85efe43293e46adc0eff5fc+IMAGE_TINY+IMAGE_TINY.png
Figure 7.15.9
  1. k=2, center is D
f-d_108b68250a18fce762d94730f47952cd2cf24462d72ddf62232bdfdc+IMAGE_TINY+IMAGE_TINY.png
Figure 7.15.10
  1. k=34, center is A
f-d_1ab555015b90eb37e7cc4bd9035d20e0912e7c39c989f31d3dc91546+IMAGE_TINY+IMAGE_TINY.png
Figure 7.15.11
  1. k=25, center is A\)
f-d_6500b50fe4c5a43981651b9c73eedb302835a029f489427b36cb28b4+IMAGE_TINY+IMAGE_TINY.png
Figure 7.15.12

In the four questions below, you are told the scale factor. Determine the dimensions of the dilation. In each diagram, the black figure is the original and P is the center of dilation.

  1. k=4
f-d_02235f9cb7eed8859781462896c6a07ae771af5fa9e07b71414b98a6+IMAGE_TINY+IMAGE_TINY.png
Figure 7.15.13
  1. k=13
f-d_1894d916a18925db05c0c57b81ef99e240e87c0ca40eb929b4e73177+IMAGE_TINY+IMAGE_TINY.png
Figure 7.15.14
  1. k=2.5
f-d_b2b9fa8004034eaf0b5a3485c960d2389725e08abcc66357bb0a5fec+IMAGE_TINY+IMAGE_TINY.png
Figure 7.15.15
  1. k=14
f-d_1bd392be823bdb1d8daaf8bf4a52d340d710df7f0bdc2bfebb5b0648+IMAGE_TINY+IMAGE_TINY.png
Figure 7.15.16

In the three questions below, find the scale factor, given the corresponding sides. In each diagram, the black figure is the original and P is the center of dilation.

  1. f-d_c18591f06e677df57a1525315d06b1cc18675bb98a861e46e055e55b+IMAGE_TINY+IMAGE_TINY.png
    Figure 7.15.17
  2. f-d_65f303d08f54846cc0e370dc11878fe0cbc93432f474a7e39379652a+IMAGE_TINY+IMAGE_TINY.png
    Figure 7.15.18
  3. f-d_e243765457f99a2c482c54227a72649663b6e419c9c60fb4855aaa75+IMAGE_TINY+IMAGE_TINY.png
    Figure 7.15.19

Review (Answers)

To see the Review answers, open this PDF file and look for section 7.11.

Vocabulary

Term Definition
Dilation To reduce or enlarge a figure according to a scale factor is a dilation.
Quadrilateral 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.

Additional Resources

Interactive Element

Video: Dilation Principles - Basic

Activities: Dilation Discussion Questions

Study Aids: Types of Transformations Study Guide

Practice: Dilation of a Shape

Real World: The CSI Effect


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