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Rotations

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How to rotate points and shapes on a grid by a quarter turn about the origin or a half turn about any point, using tracing paper or counting squares, and how to find the centre and describe a rotation fully.

What a rotation is

A rotation turns a shape about a fixed point, called the centre of rotation. The image is exactly the same size and shape as the original. It has only been turned, not flipped over.

In the exam you are usually asked to rotate a shape on a grid, for 2 marks: for example 90^\circ clockwise about (0,0), or 180^\circ about a given point. You may also be asked to describe fully the single transformation that maps one shape onto another, for 2 marks.

Every rotation has three facts: the angle, the direction and the centre. A quarter turn is 90^\circ and a half turn is 180^\circ. Clockwise is the way the hands of a clock turn, and anticlockwise is the other way.

Here triangle P is turned 90^\circ clockwise about the origin O to give triangle Q. The whole triangle swings round O, like a door swinging on its hinge.

Rotating with tracing paper

You can ask for tracing paper in the exam, and it makes rotations much easier.

  • Trace the shape, and mark the centre of rotation on the tracing paper.
  • Draw a short arrow on the tracing paper, pointing straight up from the centre.
  • Hold your pencil point on the centre and turn the paper. For 90^\circ clockwise, turn until the arrow points right. For 90^\circ anticlockwise, turn until it points left. For 180^\circ, turn until it points down.
  • Mark where each corner lands, then join the corners with a ruler.

A half turn ends in the same place whichever way you turn, so a rotation of 180^\circ does not need a direction.

Counting squares from the centre

You can also rotate a point by counting squares. Write down the journey from the centre to the point, such as 3 right and 1 up. When the point turns, its journey turns with it.

  • Half turn, 180^\circ: every direction reverses. Right becomes left, and up becomes down.
  • 90^\circ clockwise: right becomes down, down becomes left, left becomes up, and up becomes right.
  • 90^\circ anticlockwise: right becomes up, up becomes left, left becomes down, and down becomes right.

Here P is 3 right and 1 up from the origin O. After 90^\circ clockwise, 3 right becomes 3 down, and 1 up becomes 1 right. So P′ is 3 down and 1 right from O.

The image of a point is written with a small dash, called a prime: the image of P is P′, read "P prime". A point and its image are always the same distance from the centre, and the centre itself does not move.

Worked example 1: A half turn about the origin

The point M is at (4,2). M is rotated 180^\circ about the origin O. Write down the coordinates of M′.

  1. From O, M is 4 right and 2 up.
  2. A half turn reverses every direction, so M′ is 4 left and 2 down from O.

  3. Your turn. Write down the coordinates of M′.

    Write your answer as (x, y)

    4 left and 2 down from O is the point (-4,-2)

  4. Answer: M′(-4,-2)

Quick rules about the origin

When the centre is the origin O, counting squares gives three quick rules. Use them, then check on the grid.

  • 180^\circ: change both signs. (5,1) goes to (-5,-1).
  • 90^\circ clockwise: swap the coordinates, then change the sign of the second one. (3,2) goes to (2,-3).
  • 90^\circ anticlockwise: swap the coordinates, then change the sign of the first one. (3,2) goes to (-2,3).

To check a quarter turn, the point and its image should be the same distance from O, with a right angle between them at O. To check a half turn, O should be exactly halfway between them.

Worked example 2: Turning a triangle 90° clockwise

Triangle DEF has vertices D(1,1), E(4,2) and F(3,4). Rotate triangle DEF 90^\circ clockwise about O.

  1. Rotate one corner at a time. Clockwise, right becomes down and up becomes right.
  2. D is 1 right and 1 up from O, so D′ is 1 down and 1 right: D′(1,-1)
  3. E is 4 right and 2 up from O, so E′ is 4 down and 2 right: E′(2,-4)

  4. Your turn. F is 3 right and 4 up from O. Write down the coordinates of F′.

    Write your answer as (x, y)

    3 right becomes 3 down, and 4 up becomes 4 right: F′(4,-3)

  5. Check with the quick rule: swap (3,4) to get (4,3), then change the sign of the second one to get (4,-3)
  6. Answer: D′(1,-1), E′(2,-4) and F′(4,-3)

When the centre is not the origin

On past papers, a centre that is not the origin has come with a half turn, 180^\circ. Count each journey from that centre, such as (4,3), not from O. Reverse the journey, then count it out again from the centre.

Mark the centre clearly on the grid before you start. A common error in the exam is turning the shape the right amount but about the wrong point, which puts the image in the wrong place.

Tracing paper works in the same way: hold your pencil point on the centre given in the question.

Worked example 3: A half turn about a point

Triangle PQR has vertices P(5,3), Q(7,3) and R(5,1). Rotate triangle PQR 180^\circ about the point C(4,3).

  1. Mark C(4,3) on the grid. A half turn reverses every journey from C.
  2. P is 1 right of C, so P′ is 1 left of C: P′(3,3)
  3. Q is 3 right of C, so Q′ is 3 left of C: Q′(1,3)

  4. Your turn. R is 1 right and 2 down from C. Write down the coordinates of R′.

    Write your answer as (x, y)

    R′ is 1 left and 2 up from C: R′(3,5)

  5. Answer: P′(3,3), Q′(1,3) and R′(3,5)

Worked example 4: A quarter turn anticlockwise

Parallelogram WXYZ has vertices W(-4,3), X(-1,3), Y(-2,5) and Z(-5,5). Rotate parallelogram WXYZ 90^\circ anticlockwise about O.

  1. Rotate one corner at a time, counting from O. Anticlockwise, left becomes down and up becomes left.
  2. W is 4 left and 3 up from O, so W′ is 4 down and 3 left: W′(-3,-4)
  3. X is 1 left and 3 up from O, so X′ is 1 down and 3 left: X′(-3,-1)

  4. Your turn. Y is 2 left and 5 up from O. Write down the coordinates of Y′.

    Write your answer as (x, y)

    2 left becomes 2 down, and 5 up becomes 5 left: Y′(-5,-2)

  5. Z is 5 left and 5 up from O, so Z′ is 5 down and 5 left: Z′(-5,-5)
  6. Answer: W′(-3,-4), X′(-3,-1), Y′(-5,-2) and Z′(-5,-5)

Describing a rotation fully

To describe fully a rotation, give the word rotation, the angle, the direction and the centre, written as coordinates. For example: rotation 90^\circ clockwise about (0,0). For 180^\circ you can leave out the direction.

Describe the turn from the first shape named in the question to the second, not the other way round. Use the word rotation, not turn or spin, and give only one transformation.

To find the centre of a half turn, pick a corner and its matching corner on the image. The centre is exactly halfway between them: halfway between the x-coordinates, and halfway between the y-coordinates.

For a quarter turn, the centre on past papers has been the origin O. Check with tracing paper: hold your pencil on O, turn the paper 90^\circ, and see whether the tracing lands on the image.

Worked example 5: Describing a quarter turn

Describe fully the single transformation that maps triangle S onto triangle T.

  1. T is the same size and shape as S, and a tracing of S fits onto T without turning the paper over. So the transformation is a rotation.
  2. The side of length 3 is across in S but up and down in T, so this is a quarter turn: 90^\circ.
  3. Try the centre O, turning clockwise. The corner (5,-1) is 5 right and 1 down from O, so it goes to 5 down and 1 left: (-1,-5), a corner of T.

  4. Your turn. Check another corner. Where does (5,-3) go after a 90^\circ clockwise turn about O?

    Write your answer as (x, y)

    5 right and 3 down becomes 5 down and 3 left: (-3,-5), another corner of T

  5. The last corner, (2,-1), goes to (-1,-2), which is also a corner of T. Every corner fits.
  6. Answer: rotation 90^\circ clockwise about (0,0)

Worked example 6: Describing a half turn

Describe fully the single transformation that maps triangle S onto triangle T.

  1. T is the same size and shape as S, but upside down, and a tracing fits without turning the paper over. So it is a rotation of 180^\circ, and no direction is needed.
  2. Match the corners: the right angle of S at (5,1) goes to the right angle of T at (-3,-3).

  3. Your turn. The centre is halfway between them. Work out the point halfway between (5,1) and (-3,-3).

    Write your answer as (x, y)

    5 and -3 are 8 apart, so halfway is 5-4=1. 1 and -3 are 4 apart, so halfway is 1-2=-1. The centre is (1,-1)

  4. Check with another pair: (4,1) and (-2,-3) also have (1,-1) halfway between them.
  5. Answer: rotation 180^\circ about (1,-1)

Common mistakes

  • Turning about the wrong centre. Count from the centre given in the question, and keep your pencil on it when you use tracing paper.
  • Turning the wrong way. Clockwise is the way a clock's hands go, and a description goes from the first shape to the second.
  • Reflecting or sliding the shape instead. A rotation turns the shape, so it must not be flipped over or just moved across.
  • Leaving out the angle or the centre. Describe a rotation with the word rotation, the angle, the direction for 90^\circ, and the centre.
  • Writing the centre the wrong way round. Coordinates go across first, then up: (-1,0) is not the same point as (0,-1).
  • Using the wrong word or two transformations. Write rotation, not turn, spin or flip, and describe only one transformation.

Practice questions

More on this topic: Rotations worksheet with full solutions