1. 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?

2. 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.

3. 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’

4. 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.

5. Dilations Scale factor - The amount of change written as a ratio
To find the new coordinates, multiply the original coordinates by the scale factor.

6. Steps to Follow Plot the given points.
Multiply each coordinate by the scale factor.
Plot the image points.
State the coordinates of the dilation.

7. 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.

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

9. 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)

10. 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)_______________________

11. 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

12. 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

13. 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

14. 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

15. Construction: How to dilate objectives?

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

17. 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

18. 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

19. 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

20. 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

21. 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