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!

Rotations Rotate the shape T anticlockwise center T T T T y x 1 2 3 4 5 6 7 - 8 x y T Rotate the shape T anticlockwise center T T T

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

Rotations Rotate the shape T anticlockwise center T T T T T T T y x 1 2 3 4 5 6 7 - 8 x y T Rotate the shape T anticlockwise center T T T - 5 T T T

Rotations Rotate the shape T clockwise center T T T T y x 1 2 3 4 5 6 7 - 8 x y T T T Rotate the shape T clockwise center T

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.

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.

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

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

Rotations Rotate the shape T anticlockwise centre 1 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?

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.

Rotations Rotate the shape T anticlockwise center 1 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?

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.

Rotations Rotate the shape T clockwise center 1 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?

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.

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”’

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)

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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

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)

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

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

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.