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Rotations

FoundationHigherAQAEdexcelOCR

Rotations is a key geometry topic at GCSE Maths. This Foundation and Higher worksheet gives you exam-style questions on rotating shapes about a point, with a full mark scheme so you can check every method mark. Suitable for AQA, Edexcel and OCR. Download the free PDF and answers below. Give the centre, angle and direction to fully describe a rotation.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

A rotation turns a shape about a fixed point called the centre of rotation. The shape keeps its size and shape, so the image is congruent to the object.

A full description needs three things: the angle of turn, the direction, and the centre of rotation. Missing any one of them loses a mark, and the centre is the part most often forgotten.

A rotation of \(180^\circ\) needs no direction, since turning clockwise and anticlockwise give the same result. For any other angle, clockwise or anticlockwise must be stated.

Revision notes

Performing a rotation

Use tracing paper: trace the shape, hold a pencil on the centre, and turn the paper by the required angle.

This is the most reliable method in an exam, and tracing paper is normally provided.

The three parts of a description

State the angle, the direction, and the centre of rotation.

For example: a rotation of \(90^\circ\) clockwise about \((0, 0)\). All three parts are needed for full marks.

Finding the centre

Join matching vertices on the object and image, then construct the perpendicular bisector of each line.

The centre of rotation is where those bisectors cross. Trial and error with tracing paper also works and is often quicker.

Key points

  • A rotation turns a shape about a centre.
  • The object and image are congruent.
  • A description needs angle, direction and centre.
  • \(180^\circ\) needs no direction.
  • Use tracing paper to perform a rotation.
  • Find the centre using perpendicular bisectors.

Worked examples

Example 1

Rotate the point \((3, 0)\) by \(180^\circ\) about the origin.

Working

\[\text{A } 180^\circ \text{ turn reverses both signs}\]apply the rule
\[(-3, 0)\]state the image point

Example 2

Rotate the point \((2, 5)\) by \(90^\circ\) clockwise about the origin.

Working

\[(x, y) \to (y, -x)\]apply the 90 degree clockwise rule
\[(5, -2)\]state the image point

Example 3

State the three things needed to describe a rotation fully.

Working

\[\text{The angle of rotation}\]how far it turns
\[\text{The direction}\]clockwise or anticlockwise
\[\text{The centre of rotation}\]the fixed point it turns about

Common mistakes

  • Forgetting the centre of rotation.

    It is the most commonly omitted part of a description, and it costs a mark.

  • Giving a direction for a 180 degree turn.

    It is unnecessary, since both directions give the same image.

  • Rotating about the wrong point.

    Read the centre carefully — it is often not the origin.

  • Confusing clockwise and anticlockwise.

    Clockwise follows the direction of clock hands.

Exam tips

  • Use tracing paper — it is the most reliable method.
  • State all three parts of the description every time.
  • Check the image is the same size and shape as the object.
  • Find the centre with perpendicular bisectors if it is not given.

Key terms

Rotation
A turn about a fixed centre.
Centre of rotation
The fixed point a shape turns about.
Anticlockwise
In the opposite direction to clock hands.
Congruent
Identical in shape and size.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.