9.7 Dilations

Scale factor—can be written as a percent, a decimal, or a fraction. The

proportion that a representation of an object bears to the object itself. It is the multiplier used to determine the size of the image.

The ratio of the corresponding segments of the image and original figure after a dilation is equal to the scale factor. The numerator is the length of a segment in the image. The denominator of the ratio is the length of a corresponding segment in the original figure.

Dilation—A dilation is a transformation that produces an image that is the same shape as the original, but is a different size. A dilation stretches or shrinks the original figure.

The dilation is transformation with center O and positive scale factor K in which every point P has an image P’ on ray OP so that OP’ = k * OP.

If k > 1, the dilation is an enlargement (the image is bigger than the original figure).

If k = 1, the original figure and its image are congruent.

If 0 < k < 1, the dilation is a reduction (the image is smaller than the original figure).

To find the center of an image and the figure after a dilation, draw a line through each pair of corresponding vertices—A and A`, B and B`, etc, Where the lines intersect, is the center. The center can be a vertex of both figures.

Dilations (unless k = 1) do not preserve congruence. This is because the corresponding lengths are not the same.

Properties of dilations

Properties preserved under a dilation:
1. angle measures (remain the same)
2. parallelism (parallel lines remain parallel)
3. colinearity (points stay on the same lines)
4. midpoint (midpoints remain the same in each figure)
5. orientation (lettering order remains the same)

PROBLEM: Draw the dilation image of triangle ABC with the center of dilation at the origin and a scale factor of 2.

OBSERVE: Notice how EVERY coordinate of the original triangle has been multiplied by the scale factor (x2).

HINT: Dilations involve multiplication!

PROBLEM: Draw the dilation image of pentagon ABCDE with the center of dilation at the origin and a scale factor of 1/3.

OBSERVE: Notice how EVERY coordinate of the original pentagon has been multiplied by the scale factor (1/3).

HINT: Multiplying by 1/3 is the same as dividing by 3!

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