Rotations

Imagine turning the page with your pencil point on O. Which way do the hands of a clock turn?

A half turn about O takes a point to the opposite side of O, the same distance away.

Use tracing paper, or think of the line from O to B turning a quarter turn clockwise.

Anticlockwise is the opposite way to the hands of a clock. Think of the line from O to C turning a quarter turn that way.

Each vertex goes to the opposite side of O, the same distance away.

Think of the line from O to D turning a quarter turn the same way as the hands of a clock.

1. To get from O to D you go 2 left and 3 up.
2. After a quarter turn clockwise, "up" becomes "right" and "left" becomes "up": 3 right and 2 up.
3. The image of D is (3,2).

The centre of rotation is the one point that does not move. Is there a corner that A and B share?

1. The centre of rotation is the only point that stays in the same place.
2. Triangles A and B share the corner at (1,2). Turning A a quarter turn clockwise about this corner gives B.
3. The centre of rotation is (1,2).

E is straight below O. Which way does a clock hand pointing down turn next?

1. E is 3 units straight down from O.
2. A clock hand pointing straight down turns to point left after a quarter turn clockwise.
3. So the image is 3 units to the left of O, at (-3,0).

Rotate one vertex at a time. Count how far right or left and up or down each one is from O.

Part (a)

1. P is at (-4,1): 4 left and 1 up from O.
2. A quarter turn clockwise makes "up" into "right" and "left" into "up": 1 right and 4 up.
3. P′ is (1,4).

Part (b)

1. Q is at (-2,1): 2 left and 1 up from O.
2. After the quarter turn clockwise it is 1 right and 2 up.
3. Q′ is (1,2).

Part (c)

1. R is at (-2,4): 2 left and 4 up from O.
2. After the quarter turn clockwise it is 4 right and 2 up.
3. R′ is (4,2).

Use tracing paper: put your pencil on a possible centre, turn the tracing a quarter turn each way and see where A lands.

1. Shape A is below the x-axis and shape B is above it, on the same side of the y-axis, so it is a quarter turn, not a half turn.
2. A is in the bottom-right of the grid and B is in the top-right. Turning about O from bottom-right to top-right is a quarter turn the opposite way to a clock's hands: anticlockwise.
3. Check: a quarter turn anticlockwise about O sends (1,-1) to (1,1) and (4,-3) to (3,4), which are corners of B.
4. Check the centre: turning about (1,1) would send (1,-1) to (3,1) and (3,-1) to (3,3), which is not shape B.
5. The rotation is 90° anticlockwise about (0,0).

Count how far F is from C, not from O. Then turn that move a quarter turn anticlockwise.

1. To get from C to F you go 2 right and 2 up.
2. After a quarter turn anticlockwise, "right" becomes "up" and "up" becomes "left": 2 left and 2 up from C.
3. From C(1,0), 2 left and 2 up is (-1,2).

For a half turn, each vertex ends up on the opposite side of C, the same distance away.

Part (a)

1. From C(-1,2) to D(1,1) is 2 right and 1 down.
2. A half turn reverses this: 2 left and 1 up from C.
3. D′ is (-3,3).

Part (b)

1. From C to E(3,1) is 4 right and 1 down.
2. A half turn makes this 4 left and 1 up from C.
3. E′ is (-5,3).

Part (c)

1. From C to F(3,3) is 4 right and 1 up.
2. A half turn makes this 4 left and 1 down from C.
3. F′ is (-5,1).

Work backwards: undo a quarter turn clockwise by turning a quarter turn anticlockwise.

1. To get back to G, turn the image a quarter turn anticlockwise about O.
2. The image is 2 right and 5 down from O. After a quarter turn anticlockwise, "right" becomes "up" and "down" becomes "right": 5 right and 2 up.
3. G is (5,2). Check: turning (5,2) a quarter turn clockwise gives (2,-5).

In a half turn the centre is exactly halfway between a corner and its image. Match the right-angled corners first.

1. The right-angled corner of A is (1,1). The right-angled corner of B is (-1,-3).
2. In a half turn the centre is halfway between a point and its image. Halfway between x=1 and x=-1 is 0, and halfway between y=1 and y=-3 is -1.
3. Check with another pair: (4,1) and (-4,-3) also have (0,-1) halfway between them.
4. The centre of rotation is (0,-1).

Is R turned the right amount and the right way? If so, is it in the right place?

1. The side of T that is 3 units long goes up from (1,2). In R the matching side goes right from (2,1). "Up" has become "right", so R has been turned a quarter turn clockwise: the angle and direction are right.
2. Rotating about (0,0) would send the corner (1,2) to (2,-1). In R that corner is at (2,1), two squares too high.
3. R is the right shape turned the right way, but in the wrong place, so Ravi used the wrong centre of rotation.

In a half turn, C is exactly halfway between any point and its image.

Part (a)

1. C is halfway between (3,5) and (-1,-3).
2. Halfway between x=3 and x=-1 is 1. Halfway between y=5 and y=-3 is 1.
3. C is (1,1).

Part (b)

1. From C(1,1) to (4,-2) is 3 right and 3 down.
2. A half turn reverses this: 3 left and 3 up from C.
3. The image is (-2,4).

Use A and A′, and B and B′, to work out the angle and direction of the turn. Then turn C and D the same way about O.

Part (a)

1. A(1,1) goes to A′(-1,1) and B(4,1) goes to B′(-1,4). AB points right and A′B′ points up, so the rotation is a quarter turn anticlockwise: 90° anticlockwise about O.
2. C is at (4,3): 4 right and 3 up from O. After a quarter turn anticlockwise, "right" becomes "up" and "up" becomes "left": 3 left and 4 up.
3. C′ is (-3,4).

Part (b)

1. D is at (2,4): 2 right and 4 up from O.
2. After a quarter turn anticlockwise it is 4 left and 2 up.
3. D′ is (-4,2).

Rotating R′ through 180° about C takes it back to R.

1. R′ is at (-1,1). A half turn about C undoes itself, so turning R′ through 180° about C gives R.
2. From C(1,2) to R′(-1,1) is 2 left and 1 down. The opposite move from C is 2 right and 1 up.
3. R is (3,3). Check: R′ is 2 squares above P′, so after a half turn R is 2 squares below P(3,5).

Rotate each of A, B and C yourself and compare with Mia's answers.

Part (a)

1. A(1,2) is 1 right, 2 up. A quarter turn clockwise gives 2 right, 1 down: (2,-1). Mia is right.
2. B(4,2) gives 2 right, 4 down: (2,-4). Mia is right.
3. C(1,5) gives 5 right, 1 down: (5,-1), not (1,-5). C′ is wrong.

Part (b)

1. C is at (1,5): 1 right and 5 up from O.
2. After a quarter turn clockwise, "up" becomes "right" and "right" becomes "down": 5 right and 1 down.
3. The correct C′ is (5,-1). Mia only changed the sign of the y-coordinate, which is a reflection in the x-axis.

Find the angle, the direction and the centre. Tracing paper helps: try turning about O first.

Part (a)

1. The right-angled corner of A is (-2,1). From there one side goes left to (-4,1) and the other goes up to (-2,3).
2. In B the right-angled corner is (1,2). One side goes up to (1,4) and the other goes right to (3,2). "Left" has become "up" and "up" has become "right": a quarter turn clockwise.
3. About O, (-2,1) goes to (1,2) and (-4,1) goes to (1,4), which matches B. About (-1,1), (-4,1) would go to (-1,4), which is not on B.
4. The rotation is 90° clockwise about (0,0).

Part (b)

1. (-4,-1) is 4 left and 1 down from O.
2. After a quarter turn clockwise, "left" becomes "up" and "down" becomes "left": 1 left and 4 up.
3. The image is (-1,4).