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10.3 Rotation

GCSE Maths Foundation · Edexcel · Chapter 10: Transformations

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: turning shapes about a centre, and describing a rotation fully. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Rotate a shape through 90° or 180° about the origin or another centre
  • Use tracing paper to carry out a rotation
  • Find the image of a point after a rotation about the origin
  • Describe a rotation fully: angle, direction and centre

Key words

  • Rotation: A turn about a fixed point. The image is congruent to the object.
  • Centre of rotation: The fixed point the shape turns about. It does not move.
  • Clockwise: The way the hands of a clock turn. Anticlockwise is the opposite way.
  • Angle of rotation: How far the shape turns: usually 90° (a quarter turn) or 180° (a half turn).

Three facts describe a rotation

To describe a rotation fully you need the angle, the direction (clockwise or anticlockwise) and the centre. For a 180° turn the direction does not matter.

Key fact: 90° clockwise is the same as 270° anticlockwise. 180° clockwise is the same as 180° anticlockwise.

Using tracing paper

  • Trace the shape and mark the centre of rotation with a cross.
  • Put your pencil point on the centre to hold the tracing paper still.
  • Turn the tracing paper through the angle in the right direction. A vertical line drawn on the tracing paper helps you see a quarter turn.
  • Mark where the corners are now, remove the tracing paper and draw the image.

Rotating about the origin

For rotations about (0, 0) there are quick rules. Always check with a sketch.

  • 90° clockwise; (x, y) goes to (y, −x); Example: (4, 1) goes to (1, −4)
  • 90° anticlockwise; (x, y) goes to (−y, x); Example: (4, 1) goes to (−1, 4)
  • 180°; (x, y) goes to (−x, −y); Example: (4, 1) goes to (−4, −1)

Worked example: a triangle about the origin

Triangle A has vertices (1, 1), (4, 1) and (1, 3). Rotate it 90° clockwise about (0, 0).
(1, 1) → (1, −1). (4, 1) → (1, −4). (1, 3) → (3, −1).

Answer: B has vertices (1, −1), (1, −4) and (3, −1)

Rotating about another point

Use tracing paper, or count from the centre. Find how far each vertex is from the centre, then turn that “step” through the angle.

Worked example: counting from the centre

Rotate the vertex (5, 1) of T 90° anticlockwise about (2, 1).
From the centre, (5, 1) is 3 right. A quarter turn anticlockwise turns “3 right” into “3 up”.
So the image is 3 up from (2, 1).

Answer: (2, 4)

Worked example: a half turn

Rotate (4, 2) through 180° about (2, 1).
From the centre, (4, 2) is 2 right and 1 up. A half turn makes this 2 left and 1 down: (2 − 2, 1 − 1).

Answer: (0, 0)

Common mistakes

  • Leaving out part of the description. You need the angle, the direction (unless it is 180°) and the centre.
  • Turning the wrong way. Clockwise is the way clock hands move.
  • Rotating about the wrong point, e.g. about a corner of the shape instead of the given centre.
  • Mixing up rotation and reflection. A rotated shape is turned, not flipped: it is still the same way round.

Revision questions

Try these, then check your answers below each one.

Question 1. The point (3, 1) is rotated 90° clockwise about the origin. Write down the coordinates of its image.

Answer: (1,-3)

Working: (3, 1) → (1, -3).

Question 2. The point (−2, 5) is rotated 180° about the origin. Write down the coordinates of its image.

Answer: (2,-5)

Working: (x, y) → (-x, -y), so (-2, 5) → (2, -5).

Question 3. The point (4, 2) is rotated 90° anticlockwise about the origin. Write down the coordinates of its image.

Answer: (-2,4)

Working: (4, 2) → (-2, 4).

Related lessons

Continue in the full course

This lesson is section 10.3 (EGCMF1003) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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