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Maths · Transformations and Similarity

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Rotations

Rotating shapes about a centre, describing a rotation fully, and finding the centre of a rotation.

  • 9 key terms
  • All boards

Learning Objectives

  1. 1Rotate a shape about a centre through a given angle and direction.
  2. 2Describe a rotation fully by its centre, angle and direction.
  3. 3Find the centre of a rotation using tracing paper or by reasoning.
  4. 4Use the 90 degree and 180 degree patterns for the coordinates of rotated points.

Turning a shape

A rotation turns every point of a shape through the same angle about a fixed point called the centre of rotation. The shape keeps its size and its shape, so the image is congruent to the object, but it faces a different way unless the turn is a full circle. A rotation needs three details: the centre, the angle and the direction. Quarter turns and half turns are the common ones in an exam, and most questions give you tracing paper on the practical papers or ask for coordinates only, so you need both the drawing skill and the pattern for the numbers.

Three details of a rotation

A rotation is fully described by its centre, its angle and its direction.

  • Centre

    The point that stays fixed, such as the origin or the point \((2, 1)\).

  • Angle

    Usually \(90^\circ\), \(180^\circ\) or \(270^\circ\) in a non-calculator exam.

  • Direction

    Clockwise or anticlockwise. A turn of \(180^\circ\) is the same in either direction, so the direction is not needed.

  • Distance from the centre

    Each point stays the same distance from the centre, so it moves along part of a circle.

Rotating a point about the origin

Rotate the point \((2, 5)\) through \(90^\circ\) anticlockwise about the origin, and then \((2, 5)\) through \(180^\circ\) about the origin.

Show the solutionHide the solution
  1. 1 Anticlockwise rule A \(90^\circ\) anticlockwise turn sends \((x, y)\) to \((-y, x)\).
  2. 2 Apply it \((2, 5)\) goes to \((-5, 2)\).
  3. 3 Half turn rule A \(180^\circ\) turn sends \((x, y)\) to \((-x, -y)\).
  4. 4 Apply it \((2, 5)\) goes to \((-2, -5)\).

Answer\((-5, 2)\) and \((-2, -5)\)

Rotating about another centre

When the centre is not the origin, move each point round the centre rather than the origin.

  • Draw the radius

    Join the centre to a corner, then turn that line through the angle, keeping its length.

  • Use the grid

    For \(90^\circ\), swap the horizontal and vertical distances from the centre, with the right signs.

  • Example

    A point 3 right and 1 up from the centre goes 1 right and 3 down after a \(90^\circ\) clockwise turn.

  • Tracing paper

    Trace the shape, put the pencil on the centre, and turn the paper.

Describing a rotation and finding the centre

To describe a rotation you must find the centre as well as the angle and direction.

  • Angle and direction

    Compare a side of the object with the same side of the image.

  • Finding the centre

    The centre is the same distance from a point and its image, so it is on the perpendicular bisector of the line joining them. Find it for two different points and see where the lines cross.

  • Check with tracing paper

    Put the pencil on your centre and turn the paper to see that the object lands on the image.

  • Full description

    "Rotation of \(90^\circ\) clockwise about the point \((1, 0)\)" scores all three marks.

Describing a rotation

Triangle \(A\) has a corner at \((2, 1)\), and after a rotation the matching corner of triangle \(B\) is at \((2, -1)\). The triangle has turned through \(180^\circ\). Find the centre of the rotation.

Show the solutionHide the solution
  1. 1 Half turn A \(180^\circ\) rotation takes a point to the opposite side of the centre.
  2. 2 The centre is the midpoint The centre is halfway between a point and its image.
  3. 3 Calculate \(\left(\dfrac{2 + 2}{2}, \dfrac{1 + (-1)}{2}\right) = (2, 0)\).
  4. 4 Say it fully A rotation of \(180^\circ\) about the point \((2, 0)\).

Answer\((2, 0)\)

Test yourself

  1. 1

    What three details describe a rotation?

    Show answerHide answer

    The centre, the angle and the direction.

  2. 2

    What happens to \((x, y)\) in a \(180^\circ\) rotation about the origin?

    Show answerHide answer

    It goes to \((-x, -y)\).

  3. 3

    What happens to \((x, y)\) in a \(90^\circ\) clockwise rotation about the origin?

    Show answerHide answer

    It goes to \((y, -x)\).

  4. 4

    Why is no direction needed for a \(180^\circ\) rotation?

    Show answerHide answer

    A half turn is the same clockwise and anticlockwise.

  5. 5

    Is the image of a rotation congruent to the object?

    Show answerHide answer

    Yes.

Exam technique: rotations

Take care with the direction and the centre.

  • Write all three details

    Centre, angle and direction.

  • Use tracing paper

    It is allowed, and it avoids mistakes with the direction.

  • Check one point

    Turn one corner by hand and make sure it lands on the right image corner.

  • Remember the half turn

    The direction does not matter, so do not lose time deciding.

Summary and exam focus

  • A rotation turns a shape through an angle about a fixed centre, and the image is congruent to the object.
  • A full description gives the centre, the angle and the direction.
  • About the origin, a \(180^\circ\) turn sends \((x, y)\) to \((-x, -y)\) and a \(90^\circ\) clockwise turn sends it to \((y, -x)\).
  • The centre of a rotation is the same distance from a point and its image.

Exam focus

Rotate the point \((3, 1)\) through \(90^\circ\) clockwise about the origin. Write down the coordinates of the image. (2 marks) (2 marks)

A \(90^\circ\) clockwise turn sends \((x, y)\) to \((y, -x)\), so \((3, 1)\) goes to \((1, -3)\). Check by sketching a quick diagram: the point starts to the right of the \(y\)-axis and ends below the \(x\)-axis.

Key terms

The words this lesson expects you to use. Each one is linked from the first place it appears above.

Rotation
A transformation that turns a shape about a fixed point.
Centre of rotation
The fixed point about which a shape is turned.
Clockwise
The direction of the hands of a clock.
Anticlockwise
The direction opposite to the hands of a clock.
Quarter turn
A rotation of \(90^\circ\).
Half turn
A rotation of \(180^\circ\).
Angle of rotation
The size of the turn, in degrees.
Perpendicular bisector
A line that cuts another line in half at right angles.
Image
The shape after a transformation.

Questions and answers

15 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Write down 2 marks Easier

Write down the coordinates of the image of the point \((3, -2)\) after a rotation of \(180^\circ\) about the origin.

Mark scheme — 2 marks available

  • Both coordinates change sign — M1
  • \((-3, 2)\) — A1

Model answer

A \(180^\circ\) rotation about the origin changes the sign of both coordinates, so the image is \((-3, 2)\).

2. Exam question Rotate 3 marks Core

Rotate triangle \(T\) through \(90^\circ\) clockwise about the point \(O\), the origin. (3 marks)

Triangle T on a grid with the origin O marked.

Mark scheme — 3 marks available

  • Rotates at least two vertices by \(90^\circ\) about the origin — M1
  • At least two vertices correct, such as \((1, -1)\) and \((1, -3)\) — A1
  • Triangle with vertices \((1, -1)\), \((1, -3)\) and \((4, -1)\) — A1

Model answer

A \(90^\circ\) clockwise turn about the origin sends \((x, y)\) to \((y, -x)\). The vertices \((1, 1)\), \((3, 1)\) and \((1, 4)\) go to \((1, -1)\), \((1, -3)\) and \((4, -1)\).

3. Exam question Describe 3 marks Core

Describe fully the single transformation that maps triangle \(A\) onto triangle \(B\).

Triangle A and its image triangle B on a grid.

Mark scheme — 3 marks available

  • Rotation — B1
  • \(90^\circ\) anticlockwise — B1
  • About the origin, or the point (0, 0) — B1

Model answer

The point \((2, 1)\) goes to \((-1, 2)\), and \((5, 1)\) goes to \((-1, 5)\). This is the rule \((x, y) \to (-y, x)\), which is a rotation of \(90^\circ\) anticlockwise about the origin.

4. Exam question Find 2 marks Core

A rotation of \(180^\circ\) maps the point \((1, 4)\) onto the point \((5, 0)\). Find the coordinates of the centre of the rotation.

Mark scheme — 2 marks available

  • \(\dfrac{1 + 5}{2}\) or \(\dfrac{4 + 0}{2}\) — M1
  • \((3, 2)\) — A1

Model answer

The centre of a \(180^\circ\) rotation is the midpoint of a point and its image: \(\left(\dfrac{1 + 5}{2}, \dfrac{4 + 0}{2}\right) = (3, 2)\).

5. Exam question Work out 3 marks Core

The point \(P\) is \((4, 1)\). \(P\) is rotated through \(90^\circ\) clockwise about the origin to give \(Q\). \(Q\) is reflected in the \(x\)-axis to give \(R\). Write down the coordinates of \(R\).

Mark scheme — 3 marks available

  • \(Q = (1, -4)\) — B1
  • Changes the sign of the \(y\)-coordinate of their \(Q\) — M1
  • \((1, 4)\) — A1

Model answer

The rotation sends \((x, y)\) to \((y, -x)\), so \(Q = (1, -4)\). Reflecting in the \(x\)-axis changes the sign of \(y\), so \(R = (1, 4)\).

6. Exam question Work out 3 marks Stretch

The point \((3, 2)\) is rotated through \(90^\circ\) clockwise about the point \((1, 0)\). Work out the coordinates of the image.

Mark scheme — 3 marks available

  • Position relative to the centre, \((2, 2)\) — M1
  • Rotates it to \((2, -2)\) relative to the centre — M1
  • \((3, -2)\) — A1

Model answer

The point is 2 right and 2 up from the centre. A \(90^\circ\) clockwise turn sends \((x, y)\) to \((y, -x)\), so the new position is 2 right and 2 down from the centre. The image is \((1 + 2, 0 - 2) = (3, -2)\).

7. Multiple choice 1 mark Easier

What three details describe a rotation?

  1. A The mirror line and the angle
  2. B The scale factor and the centre
  3. C The centre, the angle and the direction Correct
  4. D The column vector and the angle

Why: A rotation is described by its centre, its angle and its direction.

8. Multiple choice 1 mark Easier

What is the image of \((3, -2)\) in a \(180^\circ\) rotation about the origin?

  1. A \((3, 2)\)
  2. B \((-3, 2)\) Correct
  3. C \((2, -3)\)
  4. D \((-3, -2)\)

Why: A \(180^\circ\) turn about the origin changes the sign of both coordinates.

9. Multiple choice 1 mark Core

What is the image of \((2, 5)\) in a \(90^\circ\) anticlockwise rotation about the origin?

  1. A \((-5, 2)\) Correct
  2. B \((5, -2)\)
  3. C \((-2, -5)\)
  4. D \((5, 2)\)

Why: A \(90^\circ\) anticlockwise turn sends \((x, y)\) to \((-y, x)\).

10. Multiple choice 1 mark Core

What is the image of \((3, 1)\) in a \(90^\circ\) clockwise rotation about the origin?

  1. A \((-1, 3)\)
  2. B \((-3, -1)\)
  3. C \((-1, -3)\)
  4. D \((1, -3)\) Correct

Why: A \(90^\circ\) clockwise turn sends \((x, y)\) to \((y, -x)\).

11. Multiple choice 1 mark Easier

Why is no direction needed to describe a \(180^\circ\) rotation?

  1. A The shape does not move
  2. B The direction is always clockwise
  3. C A half turn is the same in both directions Correct
  4. D The centre decides the direction

Why: A half turn clockwise ends in the same place as a half turn anticlockwise.

12. Multiple choice 1 mark Easier

Is the image of a rotation congruent to the object?

  1. A No, it is always larger
  2. B Yes, it has the same size and shape Correct
  3. C No, it is always smaller
  4. D Only for a half turn

Why: A rotation does not change lengths or angles.

13. Multiple choice 1 mark Core

What is the image of \((-2, 4)\) in a \(90^\circ\) clockwise rotation about the origin?

  1. A \((4, 2)\) Correct
  2. B \((-4, -2)\)
  3. C \((2, 4)\)
  4. D \((-4, 2)\)

Why: \((x, y)\) goes to \((y, -x)\), so \((-2, 4)\) goes to \((4, 2)\).

14. Multiple choice 1 mark Stretch

The point \((4, 3)\) is rotated through \(180^\circ\) about the point \((1, 1)\). What is the image?

  1. A \((-4, -3)\)
  2. B \((-3, -2)\)
  3. C \((2, 1)\)
  4. D \((-2, -1)\) Correct

Why: The point is 3 right and 2 up from the centre, so the image is 3 left and 2 down: \((1 - 3, 1 - 2) = (-2, -1)\).

15. Multiple choice 1 mark Stretch

A \(180^\circ\) rotation takes \((1, 5)\) to \((5, 1)\). What is the centre of rotation?

  1. A \((0, 0)\)
  2. B \((4, 4)\)
  3. C \((3, 3)\) Correct
  4. D \((6, 6)\)

Why: The centre is the midpoint of a point and its image: \(\left(\dfrac{1 + 5}{2}, \dfrac{5 + 1}{2}\right) = (3, 3)\).