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**.

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.

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

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.

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

Consider the figure shown on the coordinate grid:

a

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

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b

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

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Find the scale factor for the following dilation:

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Consider the figure on the coordinate plane shown:

a

Dilate the figure around center of dilation \left(6,7\right) with a scale factor of \dfrac{1}{2}.

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b

Compare the pre-image with its image after performing the dilation.

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Idea summary

When a figure is dilated with a center of dilation at point P by a factor of k:

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

In coordinate notation, a point dilated with respect to the origin and a scale factor of k is written as \left(x,y\right)\to\left(kx,ky\right)