1. R90 (x,y)  Rotate the point (x, y) 90 counterclockwise
ROTATIONS
A rotation is a transformation that TURNS a figure around the origin (0, 0). A rotation can move in two directions, clockwise or counterclockwise.
A COUNTERCLOCKWISE rotation is a POSITIVE rotation.
A CLOCKWISE rotation is a NEGATIVE rotation.
R90 (x,y)  Rotate the point (x, y) 90 counterclockwise
R-90 (x,y)  Rotate the point (x, y) 90 clockwise
Notice the angle between the pre-image, the origin & the image is equal to the degree of rotation!

2. Rotations Rotate the shape T anticlockwise center T T T T y x 1 2 3 45 6 7 - 8 x y T
Rotate the shape T
anticlockwise
center T T T

3. Example
Rotate ABC with points A(1,-1) B(1,-4) C(5,-4) 90° about the origin.
It doesn’t say anticlockwise or counterclockwise so you have to understand that a positive turn is a counterclockwise turn

4. Rotations Rotate the shape T anticlockwise center T T T T T T T y x 12 3 4 5 6 7 - 8 x y T
Rotate the shape T
anticlockwise
center T T T - 5 T T T

5. Rotations Rotate the shape T clockwise center T T T T y x 1 2 3 4 5 67 - 8 x y T T T
Rotate the shape T
clockwise
center T

6. Rotation
In order to rotate an object we need 3 pieces of information
Center of rotation
Angle of rotation
Direction of rotation
In order to find the image, using this information it is
best to use tracing paper.

7. Rotation Center of rotation Angle of rotation 90o
Direction of rotation
clockwise
Put a piece of tracing paper
over the drawing
Copy the object onto the
tracing paper A A’
Put a pencil on the tracing
paper – point at the centre
of rotation X
Rotate the tracing paper
by the required amount in
the specified direction
Note the end point of the object.
Remove the tracing paper and
draw the image and label it.

8. 90o c/w about the origin and label it ‘a
Rotate shape A :
90o c/w about the origin and
label it ‘a
b) 180o c/w about the origin and
label it ‘b’
c) 90o anti c/w about the origin and
label it ‘c’
d) 90o c/w about the (2,2) and
label it ‘d’
e) 90o anti c/w about the (-2,1) and
label it ‘e’
f) 90o anti c/w about the (-4,6) and
label it ‘f’
g) 90o anti c/w about the (1,8) and
label it ‘g’ A 3 4 5 6 7 8 9 10 -9 -8 -7 -6 -5 -4 -3
-10 1 2 -2 -1

9. Now do these Rotation y-axis x-axis Rotate A :
90o c/w about the origin
and label it ‘a’
b) 180o c/w about the origin
and label it ‘b’
90o counter c/w about the
Origin and label it ‘c’
d) 90o c/w about the (2,2)
and label it ‘d’
e) 90o counter c/w about the (-2,1)
and label it ‘e’
f) 90o counter c/w about the (-4,6)
and label it ‘f’
g) 90o counter c/w about the (1,8)
and label it ‘g’
y-axis 1 2 3 4 5 6 7 8 9 10
x-axis -4 -5 -6 -7 -8 -9
-10 -3 -2 -1 x f d x A g a x x e x c b

10. Rotations Rotate the shape T anticlockwise centre1 2 3 4 5 6 7 - 8 x y T
Rotate the shape T
anticlockwise
centre T T T
What are the new coordinates of the triangle?
How can you work them out without drawing a grid or the shape?

11. Rotations
How to work out the coordinates without drawing a grid or the shape?
Rotate the shape T
anticlockwise center
Explain how to get from the left hand coordinate to the right hand coordinate.
Change the sign of the y coordinate and then swap the coordinates around.

12. Rotations Rotate the shape T anticlockwise center1 2 3 4 5 6 7 - 8 x y T
Rotate the shape T
anticlockwise
center T T T - 5 T T T
What are the new coordinates of the triangle?
How can you work them out without drawing a grid or the shape?

13. Rotations
How to work out the coordinates without drawing a grid or the shape?
Rotate the shape T
anticlockwise center
Explain how to get from the left hand coordinate to the right hand coordinate.
Change the sign of the x coordinate and change the sign of the y coordinate.

14. Rotations Rotate the shape T clockwise center1 2 3 4 5 6 7 - 8 x y T T T
Rotate the shape T
clockwise
center T
What are the new coordinates of the triangle?
How can you work them out without drawing a grid or the shape?

15. Rotations
How to work out the coordinates without drawing a grid or the shape?
Rotate the shape T
clockwise center
Explain how to get from the left hand coordinate to the right hand coordinate.
Just change the sign of the x coordinate and then swap the coordinates around.

16. Given ABC with A(1, 1), B(4, 1), and C(1, 3)
Rules for POSITIVE, COUNTER CLOCKWISE Rotations
Given ABC with A(1, 1), B(4, 1), and C(1, 3)
(-1,1)
(-1,4)
(-3, 1) B’
R90 (x,y)  (-y, x)
Switch and negate the 1st
R180(x, y)  (-x,-y)
Negate both
R270(x, y)  (y, -x)
Switch & Negate the 2nd C C A’ C’ A B
(-1, -1)
(-4, -1)
(-1, -3) B” A”
A”’
C”’
(1, -1)
(3, -1)
(1, -4) C”
B”’

17. ROTATION RULES
90 degrees counterclockwise around origin: (x, y) (-y, x) 180 degrees around the origin: (x, y)  (-x, -y) 270 degrees counterclockwise around origin: (x, y)  (y, -x)

18. END

19. Finding the centre of rotation Draw a line from each point
to the corresponding point
on the image. B C A
Object
Draw the perpendicular
bisector for the connecting
lines.
75o x
The centre of rotation is
where the perpendicular
bisectors cross. C’ B’ A’
75o anticlockwise
Image
The alternative (easier) method
is to trace the object onto tracing
paper and use trial and error

20. Now do these Rotation y-axis 90o c/w centre (0,0)
Find the centre of rotation for
each of these rotations
A onto B
A onto C
A onto D
D onto B
B onto F
A onto E
C onto G
H onto A
B onto E
D onto C
y-axis 1 2 3 4 5 6 7 8 9 10
x-axis -4 -5 -6 -7 -8 -9
-10 -3 -2 -1 A B F G H C D
90o c/w centre (0,0) E
180o c/w or ac/w centre (0,0)
270o c/w or 90o ac/w centre (0,0)
270o c/w or 90o ac/w centre (0,0)
90o c/w centre (2,-2)
90o ac/w centre (2,6)
90o c/w centre (-2,-1)
180o c/w or ac/w centre (0,4)
180o c/w or ac/w centre (4,2)
90o ac/w centre (0,0)

22. Worksheet 1 Rotation A Rotate shape A : 90o c/w about the origin
and label it ‘a’
b) 180o c/w about the origin
and label it ‘b’
c) 90o anti c/w about the origin
and label it ‘c’
d) 90o c/w about the (2,2)
and label it ‘d’
e) 90o anti c/w about the (-2,1)
and label it ‘e’
f) 90o anti c/w about the (-4,6)
and label it ‘f’
g) 90o anti c/w about the (1,8)
and label it ‘g’
Find the centre of rotation for
each of these rotations
A onto B
b) A onto C
c) A onto D
d) D onto B
e) B onto F
f) A onto E
C onto G
H onto A
B onto E
D onto C A 3 4 5 6 7 8 9 10 -9 -8 -7 -6 -5 -4 -3
-10 1 2 -2 -1 3 4 5 6 7 8 9 10 -9 -8 -7 -6 -5 -4 -3
-10 1 2 -2 -1 A B F G H C D

23. Rotation
Objectives:
D Grade Rotate shapes about the origin
Describe rotations fully about the origin
Identify reflection symmetry in 3-D solids
C Grade Rotate shapes about any point
Describe rotations fully about any point
Find the centre of rotation and describe it fully

24. If you are asked to rotate and object
Rotation
Centre of rotation
Angle of rotation
30o
Direction of rotation
clockwise
If you are asked to rotate and object
by an angle that you have to measure
follow the same steps and:
30o A’ A
Mark a line from the centre of rotation
to use as 0o and also mark this on the
same place on the tracing paper. X
Before putting the tracing paper on
measure the required angle, and draw
a line accordingly.