Transformations unit, Lesson 7

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Presentation transcript:

Transformations unit, Lesson 7 DILATIONS Transformations unit, Lesson 7

DILATIONS A dilation is a transformation that produces an image that is the same shape as the original, but is a different size. The image and pre-image of a dilation are not isometric – they are similar. 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.

DILATIONS 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".

DILATIONS The washers shown in this photo illustrate the concept of dilation. The washers are the same shape, but they are different in size.

DILATIONS The description of a dilation includes the scale factor (or scalar factor)and the center of the dilation.

DILATIONS 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'.

DILATIONS If the absolute value of the scale factor is greater than 1, the image is an enlargement. If the absolute value of the scale factor is between 0 and 1, the image is a reduction.

DILATIONS 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. Remember: Dilations are enlargements (or reductions).

DILATIONS A dilation of scalar factor k whose center of dilation is the origin may be written: Dk (x, y) = (kx, ky). Dilation of the point (2,3) with a scale factor of 4 is (8,12). Dilation of the point (3,1) with a scale factor of -2 is (-6, -2). Dilation of the point (12, 6) with a scale factor of 1/3 is (4,2).

DILATIONS Properties preserved under a dilation: angle measures (remain the same) parallelism (parallel lines remain parallel) colinearity (points stay on the same lines) midpoint (midpoints remain the same in each figure) orientation (lettering order remains the same) Properties NOT preserved under dilation: distance (NOT an isometry) - lengths of segments are NOT the same in all cases except when the scale factor is 1

DILATIONS EXAMPLE 1: 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 (2). HINT: Dilations involve multiplication!

DILATIONS EXAMPLE 2: 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!

Triangle A'B'C' is the image after triangle ABC is dilated. Find the scale factor used. A' B B' C' C

What scale factor was used in the dilation of ABCD?

Practice with Dilations Is this an example of a dilation? yes no Why?

Practice with Dilations Under the dilation shown at the left, the value of the "?" will be: 1.5 3 4 How do you know?

Practice with Dilations Under a dilation, the picture shown at the left has been reduced in size. The scalar factor was: ¼ ½ 2 4 How do you know?

Practice with Dilations The length of the small train is 4 inches. Under a dilation of scale factor 2, the length of the large train is: ? 2 inches 8 inches How do you know?