dilations in the coordinate plane eq: how do you describe the properties of dilation? how can you...

Dilations in the Coordinate Plane

EQ: How do you describe the properties of dilation?

How can you describe the effect of dilation on coordinates using algebraic representation?

What is the connection between transformation and similar figure?

Dilations Dilation

A transformation that changes the size of a figure. The original figure and transformed figure are similar.Corresponding angles are sameCorresponding side lengths are not same; but proportional

Center of dilation - The point of projectionIn the coordinate plane, the center of dilation is the origin.

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

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

Two types of dilationsEnlargementReduction

Dilations Scale factor - The amount

of change written as a ratio

To find the new coordinates, multiply the original coordinates by the scale factor.

Steps to Follow1. Plot the given points.

2. Multiply each coordinate by the scale factor.

3. Plot the image points.

4. State the coordinates of the dilation.

Finding a Scale Factor

The blue triangle is a dilation image of the red triangle. Describe the dilation.

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

34

84''

XT

TX

Finding a Scale Factor

The blue quadrilateral is a dilation image of the red quadrilateral. Describe the dilation.

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

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

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

Algebraic Notation of Dilation

In a coordinate plane, dilations whose centers are the origin have the property that the pre-image P to image P’ where ‘k’ is the scale factor.

P (x, y) P’ (kx, ky)

4) Algebraic to verbal (2, 3)

a)(x, y) (2x, 2y) __________________

b)New coordinates: ( )

b) (x, y) (1/4x, 1/4y)__________________ New coordinates: ( )

c) (x, y) (2.5x, 2.5y)_____________________

New coordinates: ( )

d) (x, y) (-2y, 2x)_______________________

New coordinates: ( )

Given the vertices of the triangle, find a dilation by a scale factor of 3.

A (1,2) B (3,3) C (1,3)

y

x

C B

A

C’B’

A’A’ (3,6)

B’ (9,9)

C’ (3,9)

Given the vertices of the rectangle, find a dilation by a scale factor of 2/3.

A (-6,-3) B (-6,3) C (6,3) D (6,-3)

y

x

A

B C

D

A’ (-4,-2)B’ (-4,2)

C’ (4,2)D’ (4,-2)

A’

B’ C’

D’

DILATION

Triangle: A (-2, -4)B (3, -2)C (1, 2)D (-4,0)Color Red

Scale factor of 2

Scale factor of

1/2

Scale factor of 3

Algebraic Representation

(x, y) ( )

(x, y) ( )

(x, y) ( )

New Coordinates:

A’ ( )B’ ( )C’ ( ) D’ ( )Color Blue

A’’ ( )B’’ ( )C’’ ( )D”( )Color Green

A’’’ ( )B’’’ ( )C’’’ ( )D’’’ ( )Color Pink