8.7 Dilations Geometry

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 used to create an image larger than the original is called an enlargement. A dilation used to create an image smaller than the original is called a reduction.

You are probably familiar with the word "dilate" as it relates to the eye. "The pupils of the eye were dilated." As light hits the eye, the pupil enlarges or contracts depending upon the amount of light. This concept of enlarging and contracting is "dilating".

The description of a dilation includes the scale factor and the center of the dilation. NOTE: the scale factor is sometimes referred to as the scalar factor.

In mathematics, the dilation of an object is called its image. If the original object was labeled with letters, such as polygon ABCD, the image may be labeled with the same letters followed by a prime symbol, A'B'C'D'.

If the scale factor is greater than 1, the image is an enlargement. (numerator is larger) ex: 5/4 If the scale factor is between 0 and 1, the image is a reduction. (denominator is larger) ex: 4/5 A figure and its dilation are similar figures. The lengths of the sides must be proportional and have the same scale factor when you compare corresponding sides. The length of each side of the image is equal to the length of the corresponding side of the original figure multiplied by the scale factor. The distance from the center of the dilation to each point of the image is equal to the distance from the center of the dilation to each corresponding point of the original figure times the scale factor.

Example 1: 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!

Example 2: 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!

Remember: Dilations are enlargements (or reductions)

For this example, the center of the dilation is NOT the origin. The center of dilation is a vertex of the original figure. PROBLEM: Draw the dilation image of rectangle EFGH with the center of dilation at point E and a scale factor of 1/2. OBSERVE: Point E and its image are the same. It is important to observe the distance from the center of the dilation, E, to the other points of the figure. Notice EF = 6 and E'F' = 3. HINT: Be sure to measure distances for this problem.

Triangle GHI is dilated by a factor of 2 with center of dilation at point P. G I H

Triangle GHI is dilated by a factor of 2 with center of dilation at point P. Draw the new figure. G I H G H I G’

Triangle GHI is dilated by a factor of 2 with center of dilation at point P. Draw the new figure. G I H G H I H’

Triangle GHI is dilated by a factor of 2 with center of dilation at point P. Draw the new figure. G I H G H H I’

Triangle GHI is dilated by a factor of 2 with center of dilation at point P. Draw the new figure. G I H GG’ H I’ H’