Dilations: (Stretching/Shrinking)Dilations: (Stretching/Shrinking) Dilations use a scale factor to reduce or Dilations use a scale factor to reduce or

enlarge shapes.enlarge shapes. Every dilation has a center and a scale Every dilation has a center and a scale

factor. Most of the time it is the origin (0, 0)factor. Most of the time it is the origin (0, 0) Scale Factor: tells you how many times Scale Factor: tells you how many times

larger or smaller your image will be.larger or smaller your image will be.

The new shape and the image are similar. The new shape and the image are similar. Dilations are also called similarity Dilations are also called similarity transformations.transformations.

Finding a DilationFinding a Dilation

To find a dilation with center C and To find a dilation with center C and scale factor n, you can use the scale factor n, you can use the following two rules.following two rules.

The image C is itself (meaning C’=C)The image C is itself (meaning C’=C) For any point R, R’ is on CR and CR’ = For any point R, R’ is on CR and CR’ =

nn•CR.•CR.

How do we locate dilation images?How do we locate dilation images?

A A dilationdilation is a transformation who is a transformation who preimage and image are similar. A preimage and image are similar. A dilation dilation is not is not an isometry. an isometry.

Every dilation has a center and a Every dilation has a center and a scale factor n, n >0. The scale factor scale factor n, n >0. The scale factor describes the size change from the describes the size change from the original figure to the image.original figure to the image.

Example 1:Example 1: Quadrilateral ABCD Quadrilateral ABCD

has vertices A(-2, -has vertices A(-2, -1), B(-2, 1), C(2, 1) 1), B(-2, 1), C(2, 1) and D(1, -1). and D(1, -1).

Find the coordinates Find the coordinates of the image for the of the image for the dilation with a scale dilation with a scale factor of 2 and factor of 2 and center of dilation at center of dilation at the origin. the origin.

A

B C

A’A’

B’B’ C’C’

D

D’D’

Example 2:Example 2: F(-3, -3), O(3, 3), R(0, -3) Scale factor 1/3 F(-3, -3), O(3, 3), R(0, -3) Scale factor 1/3

F

O

R

F’

O’

R’

Example 3:Example 3: T(-1, 0), H(1, 0), I(2, -2), S(-2, -2) Scale factor 4T(-1, 0), H(1, 0), I(2, -2), S(-2, -2) Scale factor 4

T H

I

T’ H’

I’S

S’

The dilation is an The dilation is an enlargementenlargement if the if the scale factor is > 1. scale factor is > 1.

The dilation is a The dilation is a reductionreduction if the if the scale factor is scale factor is between 0 and 1.between 0 and 1.

Finding a Scale FactorFinding a Scale Factor The blue triangle is a dilation image The blue triangle is a dilation image

of the red triangle. Describe the of the red triangle. Describe the dilation.dilation.

The center is The center is XX. The image is larger than the . The image is larger than the preimage, so the dilation is an enlargement.preimage, so the dilation is an enlargement.

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XTTX

Finding a Scale FactorFinding a Scale Factor The blue quadrilateral is a dilation The blue quadrilateral is a dilation

image of the red quadrilateral. image of the red quadrilateral. Describe the dilation.Describe the dilation.

Graphing Dilation ImagesGraphing Dilation Images ∆∆PZG has vertices P(2,0), Z(-1, ½), and G (1, -PZG has vertices P(2,0), Z(-1, ½), and G (1, -

2).2).What are the coordinates of the image of P for a What are the coordinates of the image of P for a dilation with center (0,0) and scale factor 3?dilation with center (0,0) and scale factor 3?

a) (5, 3)a) (5, 3) b) (6,0) c) (2/3, 0)b) (6,0) c) (2/3, 0) d) (3, -6)d) (3, -6)

Graphing Dilation ImagesGraphing Dilation Images Solution:Solution:The scale factor is 3, so The scale factor is 3, so

use the rule:use the rule:(x, y)(x, y)(3x, 3y).(3x, 3y).P(2,0) P(2,0) P’(3•2, 3•0) or P’(3•2, 3•0) or

P’(6, 0).P’(6, 0). The correct answer is B.The correct answer is B.

What are the What are the coordinates for G’ and coordinates for G’ and Z’? Z’?