Geometry:

Dilations

We have already discussed translations, reflections and

rotations. Each of these transformations is an isometry,

which means preimage. thetocongruent is image the

In a dilation

preimage. thesimilar to is image the

A dilation centered at point C with a scale factor of k, where can be defined as follows:

1. The image of point C is itself. That is, _____

2. For any point P other than C, the ____________________________

CC '

CPkCP' and CPon is P'point CPk

NOTE: If , then the dilation is a ____________

If , then the dilation is an

____________

10 k

1k

ncontractio

expansion

Why is ?''~ QCPCPQ

?similarityfor // SSSSASAA

reflexiveby CC

CPk

CQk CQk

CQ

CPk

CP

kk

11

SAS

?

''//

Why

QPPQ

Converse CAP

Example: Under a dilation, triangle A(0,0), B(0,4), C(6,0) becomes triangle A'(0,0), B'(0,10), C'(15,0).  What is the scale factor for this dilation?

B’(0, 10)

C’(15, 0)

ACk AC' ,definition By the

615 k

6

15k

Let’s consider why this theorem is true.

ABkBA ''

''CA'

CA then,

SASby B'CA'~CAB Since

BA

AB

CAk ''

1

BA

AB

k

ABkBA ''

Example: Line segment AB with endpoints A(2, 5) and B(6, -1) lies in the coordinate plane. The segment will be dilated with a scale factor of and a center at the origin to create . What will be the length of ?

52

''BA''BA

135

4132

5

2'' BA

132526)4(1562 2222 AB

Example: Under a dilation of scale factor 3 with the center at the origin, what will be the coordinates of the image of point A(3, 4)? point B(4,1)?

What do you notice about the coordinates ofpoints A and A’ as well as B and B’ in relation to the scale factor?

)12,9('A )3,12('B

preimage. in the coordinate

ingcorrespond the times3

is image in the coordinateEach

Theorem: If the center of dilation is the origin and the scale factor is k, the coordinates of the point A’, the image of A(x, y), will be __________.),( ykxk