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Geometry Dilations

# Geometry Dilations September 8, 2015 Goals Identify Dilations Make drawings using dilations

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### Text of Geometry Dilations September 8, 2015 Goals Identify Dilations Make drawings using dilations Geometry

Dilations April 21, 2023

Goals

Identify Dilations Make drawings using dilations. April 21, 2023

Rigid Transformations

Previously studied in Chapter 7.

Rotations Translations These were isometries: The pre-image and the

image were congruent. April 21, 2023

Dilation

Dilations are non-rigid transformations. The pre-image and image are similar, but not

congruent. April 21, 2023 April 21, 2023

Dilation

Enlargement April 21, 2023

Dilation

Reduction April 21, 2023

Dilation

Center of Dilation

R

S

T

C April 21, 2023

Dilation

Center of Dilation

R

S

T

C

CR

CR2CR

R April 21, 2023

Dilation

Center of Dilation

R

S

T

C

CR

CR2CR

R

CSCS

S2CS April 21, 2023

Dilation

Center of Dilation

R

S

T

C

CR

CR2CR

R

CSCS

S2CS

CT

CT2CT T April 21, 2023

Dilation

Center of Dilation

R

S

T

C

CR

CR2CR

R

CSCS

S2CS

CT

CT2CT T

RST ~ RST April 21, 2023

Dilation Definition

A dilation with center C and scale factor k is a transformation that maps every point P to a point P’ so that the following properties are true:

1. If P is not the center point C, then the image point P’ lies on CP. The scale factor k is a positive number

such that k 1 and

2. If P is the center point C, then P = P’.

3. The dilation is a reduction if 0 < k < 1, and an enlargement if k > 1.

CP'k =

CP April 21, 2023

Dilation

Center of Dilation

R

S

T

C

CR

CR2CR

R

CSCS

S2CS

CT

CT2CT T

Enlargement

' 2 2

1

CR CR

CR CR Scale Factor April 21, 2023

Dilation

Center of Dilation

R

S

T

C

CR

CR2CR

R

CSCS

S2CS

CT

CT2CT T

RST ~ R’S’T’

Scale Factor:' ' ' ' ' ' ' ' 'CR CS CT R S S T R T

CR CS CT RS ST RT April 21, 2023

Example

F G

HK

F’ G’

H’K’

C

What type of dilation is this? Reduction April 21, 2023

Example

F G

HK

F’ G’

H’K’

C

What is the scale factor?

36 12

45

15

' ' 15

4

1

5' ' 1

3

2

36

31

F Gk

FGF K

kFK

Notice:

k < 1

Reduction April 21, 2023

Remember:

The scale factor k is

If 0 < k < 1 it’s a reduction. If k > 1 it’s an enlargement.

'CPk

CP

image segment

pre-image segment April 21, 2023

Coordinate Geometry

Use the origin (0, 0) as the center of dilation. The image of P(x, y) is P’(kx, ky). Notation: P(x, y) P’(kx, ky). Read: “P maps to P prime”

You need graph paper, a ruler, pencil. April 21, 2023

Graph ABC with A(1, 1), B(3, 6), C(5, 4).

A

B

C

Notice the origin is here April 21, 2023

Using a scale factor of k = 2, locate points A’, B’, and C’. P(x, y) P’(kx, ky).

A

B

C

A(1, 1) A’(2 1, 2 1) = A’(2, 2)

A’

B(3, 6) B’(2 3, 2 6) = B’(6, 12)

B’

C(5, 4) C’(2 5, 2 4) = C’(10, 8) C’ April 21, 2023

Draw ABC.

A

B

C

A’

B’

C’ April 21, 2023

You’re done.

A

B

C

A’

B’

C’

Notice that rays drawn from the center of dilation (the origin) through every preimage point also passes through the image point. April 21, 2023

Do this problem.

R(0, 0)

Draw RSTV with

R(0, 0)

S(6, 3)

T(0, 12)

V(6, 3) S(-6, 3)

T(0, 12)

V(6, 3) April 21, 2023

Do this problem.

R(0, 0)

Draw R’S’T’V’ using a scale factor of k = 1/3.

S(-6, 3)

T(0, 12)

V(6, 3)

R’(0, 0)

S’(-2, 1)

T’(0, 4)

V’(2, 1) April 21, 2023

Do this problem.

R(0, 0)

R’S’T’V’ is a reduction.

S(-6, 3)

T(0, 12)

V(6, 3)

R’(0, 0)

S’(-2, 1)

T’(0, 4)

V’(2, 1) April 21, 2023

Summary

A dilation creates similar figures. A dilation can be a reduction or an

enlargement. If the scale factor is less than one, it’s a

reduction, and if the scale factor is greater than one it’s an enlargement. April 21, 2023

One more time…

Scale Factor = Image Size

Pre-image Size

Scale Factor = After

Before April 21, 2023

Enlargement or Reduction?

CP = 10 and CP’ = 20 Enlargement What is the Scale Factor? 2 k = CP’/CP = 20/10 = 2 April 21, 2023

Enlargement or Reduction?

CP = 150 and CP’ = 15 Reduction What is the Scale Factor? 1/10 k = CP’/CP = 15/150 = 1/10 April 21, 2023

Enlargement or Reduction?

CP = 20 and CP’ = 18 Reduction What is the Scale Factor? 9/10 k = CP’/CP = 18/20 = 9/10 April 21, 2023

Enlargement or Reduction?

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