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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.

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Dilation

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

congruent.

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Dilation

Enlargement

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Dilation

Reduction

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Dilation

Center of Dilation

R

S

T

C

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Dilation

Center of Dilation

R

S

T

C

CR

CR2CR

R

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Dilation

Center of Dilation

R

S

T

C

CR

CR2CR

R

CSCS

S2CS

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Dilation

Center of Dilation

R

S

T

C

CR

CR2CR

R

CSCS

S2CS

CT

CT2CT T

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Dilation

Center of Dilation

R

S

T

C

CR

CR2CR

R

CSCS

S2CS

CT

CT2CT T

RST ~ RST

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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

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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

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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

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Example

F G

HK

F’ G’

H’K’

C

What type of dilation is this? Reduction

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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

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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

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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.

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

A

B

C

Notice the origin is here

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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’

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Draw ABC.

A

B

C

A’

B’

C’

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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.

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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)

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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)

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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)

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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.

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One more time…

Scale Factor = Image Size

Pre-image Size

Scale Factor = After

Before

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Enlargement or Reduction?

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

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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?

CP = 15 and CP’ = 18 Enlargement What is the Scale Factor? 6/5 k = CP’/CP = 18/15 = 6/5

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