1. ROTATIONS
LESSON 30

2. ROTATIONS
In the diagram above, corresponding points on the two figures are related. Suppose P is any point on the original figure and P’ is the corresponding point on the image figure.
We say: P maps onto P’
We write: P P’

3. MAPPING RULES
We often use a coordinate grid when we work with transformations. We use a mapping rule to describe how points and their images are related.
A mapping rule tells you what to do to the coordinates of any point on the figure to determine the coordinates of tits image.
Example of Mapping rule:
(x, y) (x + 5, y - 2)
It tells you to add 5 to the x-coordinate and to subtract 2 from the y-coordinate.

4. MAPPING RULES FOR ROTATIONS
The mapping rules for rotations depend on the position of the rotation centre, the angle of rotation, and the direction of rotation.
For Clockwise rotations: 90o clockwise = 270o counterclockwise
180o clockwise = 180o counterclockwise
270o clockwise = 90 counterclockwise

5. PROPERTIES OF A ROTATION
The image is congruent to the original figure and has the same orientation.
For a 180o rotation ( half turn), line segments that join matching points pass through the rotation centre. The rotation centre is the midpoint of each of these line segments.

6. EXAMPLE 1
Triangle DEF has vertices D(7,-2), E(5,-3), and F(3,1). This triangle is rotated 90o counterclockwise about the origin.
Use a mapping rule to determine the coordinates of ther vertices of the image triangle D’E’F’.
Draw a diagram to show triangle DEF and its image.
Suppose triangle DEF were rotated 90o clockwise about the origin. Determine the coordinates of the vertices of the image triangle D”E”F”.

7. SOLUTION The mapping rule for the rotation is (x, y) (-y, x).
To determine the coordinates of each image point, multiply the second coordinate by -1, then interchange the coordinates.
D(7,-2) D’(2,7)
E(5,-3) E’(3,5)
F(3,1) F’(-1,3)

8. SOLUTION
c) A 90o clockwise rotation is equivalent to a 270o counterclockwise rotation.
The mapping rule for the rotation is (x, y) (y, -x). To determine the coordinates of each image point, multiply the first coordinate by -1, then interchange the coordinates.
D(7,-2) D”(-2,-7)
E(5,-3) E”(-3,-5)
F(3,1) F”(1,-3)

9. Combining Transformations: We can use mapping rules to combine a rotation with a translation or a reflection.
EXAMPLE 2
The translation (x, y) (x - 6, y + 2) is applied to a polygon with vertices A(7,1), B(7,3), C(6,4), D(5,3), E(3,3), and F(3,1). Then rotation (x, y) (-y, x) is applied to the image polygon A’B’C’D’E’.
Determine the coordinates of the vertices of the final image polygon A”B”C”D”E”.

10. SOLUTION A”(-3,1) D”(-5,-1) B”(-5,1) E”(-5,-3) C”(-6,0) F”(-3,-3)
Draw polygon ABCDEF on a coordinate grid.
To apply the mapping rule (x, y) (x - 6, y + 2), we subtract 6 from the x-coordinate and add 2 to the y-coordinate of each vertex of polygon ABCDEF.
This moves it 6 units left and 2 units up to become polygon A’B’C’D’E’F’.
To apply the mapping rule (x, y) (-y, x) to the image, we multiply the second coordinate by -1, then interchange the coordinates.
This rotates the image 90o counterclockwise about the origin to become polygon A”B”C”D”E”F”.
A”(-3,1) D”(-5,-1)
B”(-5,1) E”(-5,-3)
C”(-6,0) F”(-3,-3)

11. Class Work
Copy Notes to Lesson 30
Complete Lesson 30 worksheet