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 same Corresponding side lengths are not same; but proportional Center of dilation - The point of projection In the coordinate plane, the center of dilation is the origin.

CLASSIFY DILATIONS A dilation requires a center point and a scale factor. Example: A’ CA’ = 2(CA) CB’ = 2(CB) CD’ = 2(CD) r = 2 B’ A B center C D D’

Enlargement Reduction Two types of dilations The dilation is a reduction if the scale factor is between 0 and 1. The dilation is an enlargement if the scale factor is > 1.

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 Follow Plot the given points. Multiply each coordinate by the scale factor. Plot the image points. 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.

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

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) (x, y)  (2x, 2y) __________________ New coordinates: ( ) b) (x, y)  (1/4x, 1/4y)__________________ New coordinates: ( ) c) (x, y)  (2.5x, 2.5y)_____________________ d) (x, y)  (2y, 2x)_______________________

Given the vertices of the triangle, find a dilation by a scale factor of 3. x B’ C’ A (1,2) B (3,3) C (1,3) A’ (3,6) A’ B’ (9,9) C B C’ (3,9) A

Given the vertices of the rectangle, find a dilation by a scale factor of 2/3. x A (-6,-3) B (-6,3) C (6,3) D (6,-3) A’ (-4,-2) B’ (-4,2) C’ (4,2) B C D’ (4,-2) B’ C’ A’ D’ A D

Scale factor of 2 Scale factor of 1/2 Scale factor of 3 DILATION Rectangle: 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

Reduction/Enlargement The dilation is a reduction if 0 < k < 1 and it is an enlargement if k > 1. CP’ 3 1 REDUCTION: = = CP 6 2 6

Construction: How to dilate objectives? https://www.youtube.com/watch?v=wbDktALdi_s

Example 2 Draw a Dilation Draw the dilation image of ∆JKL with center C and r = ½ K J L

Example 2 Draw a Dilation Draw the dilation image of ∆JKL with center C and r = -½ K Since 0 < |r| < 1, the dilation is a reduction J L K L’ C J’ J K’ L

Example 3 Dilations in the Coordinate Plane COORDINATE GEOMETRY Triangle ABC has vertices A(7, 10), B(4, -6), and C(-2, 3). Find the image of ∆ABC after a dilation centered at the origin with the scale factor of 2. Sketch the preimage and the image. 22 20 18 16 14 12 10 8 6 4 2 -10 -5 -2 5 10 15 -4 -6 -8 -10 -12

Example 3 Dilations in the Coordinate Plane COORDINATE GEOMETRY Triangle ABC has vertices A(7, 10), B(4, -6), and C(-2, 3). Find the image of ∆ABC after a dilation centered at the origin with the scale factor of 2. Sketch the preimage and the image. 22 20 18 16 Preimage (x, y) Image (2x, 2y) A(7, 10) A’(14, 20) B(-4, 6) B’(8, -12) C(-2, 3) C’(-4, 6) 14 12 10 8 6 4 2 -10 -5 -2 5 10 15 -4 -6 -8 -10 -12

Example 4 Identify the Scale Factor Determine the scale factor for each dilation with center C. Then determine whether the dilation is an enlargement, reduction, or congruence transformation. a. 9 8 7 6 5 4 3 2 1 A’ B’ Scale factor = image length preimage length = 6 units 3 units = 2 A B C E D E’ D’ 2 4 6 8 10 12

Example 4 Identify the Scale Factor Determine the scale factor for each dilation with center C. Then determine whether the dilation is an enlargement, reduction, or congruence transformation. a. 9 8 7 6 5 4 3 2 1 Scale factor = image length preimage length = 4 units 4 units = 1 G H C J F 2 4 6 8 10 12 The dilation is a congruence transformation