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Contents · 0% complete
  1. Learning objectives
  2. Core idea
  3. Key formulae
  4. Worked examples
  5. Explore the graph
  6. Common mistakes
  7. Exam tip
  8. Guided practice
  9. Practice questions
  10. Challenge
  11. Summary
  12. Quick quiz
  13. Same topic, different board
  14. Past paper questions
Learn8300Papers 1–3 · Foundation tierGeometry and measuresReflections, rotations and translations
Geometry and measures · Subtopic 12 of 14

Reflections, Rotations and Translations

Translations slide, reflections flip and rotations turn; all three keep a shape exactly the same size. To "describe fully" you must give every detail: the vector, the mirror line, or the centre, angle and direction.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 30 minRevision 5 · updated 2026-10-27
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Section 1 of 14

Learning objectives

  • Translate a point or shape by a column vector.
  • Reflect in horizontal, vertical and diagonal lines.
  • Rotate through 90∘90^\circ or 180∘180^\circ about any centre.
  • Describe a transformation fully.
Pause and think

Write a capital R on a piece of paper. Can you turn the paper (without flipping it over) so that the R looks like its mirror image?

Section 2 of 14

Core idea

Start hereMoves that keep shape

Slide a puzzle piece, flip it over, or turn it: it still fits the same gap. These three moves, translation, reflection and rotation, never change size or shape.

The big idea

Translation: column vector. Reflection: mirror line. Rotation: centre, angle, direction. All preserve congruence. Describe fully.

1. Describing fully

A triangle translated, reflected and rotated
−8−6−4−22468−4−2240xyA (original)translated by (0, −3)reflected in x = 0rotated 180° about O
Each image is congruent to the original. Only reflection reverses it (a mirror image).
  • Translation by (ab)\begin{pmatrix}a\\b\end{pmatrix}.
  • Reflection in the line … (e.g. y=xy=x, x=2x=2).
  • Rotation: angle, direction, centre.
y=x:(x,y)→(y,x)90∘ clockwise about O:(x,y)→(y,−x)180∘ about O:(x,y)→(−x,−y)\begin{aligned}&y=x:&(x,y)&\to(y,x)\\&90^\circ\text{ clockwise about }O:&(x,y)&\to(y,-x)\\&180^\circ\text{ about }O:&(x,y)&\to(-x,-y)\end{aligned}
Worked exampleImages of (3,1)(3,1)

Reflect in y=xy=x; rotate 90∘90^\circ clockwise about OO; rotate 180∘180^\circ about OO.

Goal
Three images.
Strategy
Rules above.
  1. (1,3)(1,3)
    Swap.
  2. (1,−3)(1,-3)
    (y,−x)(y,-x).
  3. (−3,−1)(-3,-1)
    Both negate.
Answer(1,3)(1,3), (1,−3)(1,-3), (−3,−1)(-3,-1).
✓ CheckAll at distance 10\sqrt{10} from OO ✓.
Worked exampleFinding a centre of rotation

Given a shape and its rotated image.

Goal
Centre.
Strategy
The centre is equidistant from each point and its image.

  1. Join two points to their images.

  2. Draw the perpendicular bisector of each joining line; they cross at the centre.

AnswerThe intersection of the bisectors.
✓ CheckMeasure: centre to point = centre to image ✓.
Common misconceptionIncomplete description "Rotation of 90°" loses marks: give the centre and the direction.
Pause and think

Two reflections in parallel lines give which single transformation?

Summary

  • Give every detail.
  • All three preserve size and shape.
Key words
Invariant point
A point that does not move.
HistoryThe Alhambra's symmetries. The tiled walls of the Alhambra palace in Granada, built in the 1300s, use repeated reflections, rotations and translations so skilfully that later mathematicians found examples of many of the 1717 possible types of repeating wallpaper pattern there.
Fun factAmbulance writing. "AMBULANCE" is often painted reflected on the front of the vehicle, so drivers see it the right way round in their rear-view mirror. A reflection undoes a reflection.
Section 3 of 14

Key formulae

Reflectionsy=x: (x,y)→(y,x)y=−x: (x,y)→(−y,−x)y=x:\ (x,y)\to(y,x)\qquad y=-x:\ (x,y)\to(-y,-x)
Rotations about the origin90∘ anticlockwise: (x,y)→(−y,x)180∘: (x,y)→(−x,−y)90^\circ\text{ anticlockwise}:\ (x,y)\to(-y,x)\qquad180^\circ:\ (x,y)\to(-x,-y)
TrickSwap and sign. For a 90∘90^\circ rotation about the origin, swap the coordinates, then make one negative. Anticlockwise: negate the new xx. Clockwise: negate the new yy: (x,y)→(y,−x)(x,y)\to(y,-x). Check with (1,0)(1,0): anticlockwise should send it up to (0,1)(0,1).
Section 4 of 14

Worked examples

Example 1 · Reflection in a vertical line0 / 2 steps
Reflect (5,1)(5,1) in the line x=2x=2.
    Example 2 · Rotation about another centre0 / 2 steps
    Rotate (4,3)(4,3) through 180∘180^\circ about (1,2)(1,2).
      Section 5 of 14

      Explore the graph

      A point and its images
      Loading the interactive graph…
      Try this
      1. The first point is (a,b)(a,b). The others are its reflection in y=xy=x, its 90∘90^\circ anticlockwise rotation about the origin, and its 180∘180^\circ rotation.
      2. Move (a,b)(a,b) onto the line y=xy=x. Which image now sits on top of it?
      3. Check that all four points are the same distance from the origin.

      Reflections and rotations keep distances the same.

      In the real worldComputer games. Every time a game character moves, turns or flips, the program applies a translation, rotation or reflection to the coordinates of every point in the character's shape, often millions of times a second.
      Section 6 of 14

      Common mistakes

      IncorrectDescribing a rotation by the angle only
      CorrectCentre, angle, direction
      All three are needed for full marks.
      IncorrectReflecting in x=2x=2 as if it were the yy-axis
      CorrectMeasure from the line x=2x=2
      The mirror line is not always an axis.
      IncorrectConfusing x=2x=2 and y=2y=2
      Correctx=2x=2 is vertical
      All points with xx-coordinate 22.
      IncorrectWriting "turn" or "flip"
      Correct"Rotation", "reflection"
      Use the correct names.

      "It moved"

      "Rotation 90∘90^\circ clockwise about (1,2)(1,2)"

      Section 7 of 14

      Exam tip

      Exam tipOne transformation, one name. "Describe fully the single transformation" means one word (translation, reflection, rotation or enlargement) plus its details. Giving two transformations, even if correct, scores zero.
      Section 8 of 14

      Guided practice

      SA-4024-TV-0022 marksL1 FoundationNon-calculatorTranslate & reflect
      The point (2,5)(2,5) is translated by the vector (4−3)\dbinom{4}{-3}. Find the coordinates of its image.

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      Salam Academy Original Practice · Proposed mark scheme
      Section 9 of 14

      Practice questions

      SA-4024-TV1-0042 marksL1 FoundationNon-calculatorTranslate & reflect
      A translation maps (1,4)(1,4) to (−2,6)(-2,6). Find the column vector of the translation.

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      Salam Academy Original Practice · Proposed mark scheme
      SA-4024-TV1-0012 marksL1 FoundationNon-calculatorTranslate & reflect
      The point (2,5)(2,5) is reflected in the xx-axis. Find its image.

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      Salam Academy Original Practice · Proposed mark scheme
      SA-4024-TV-0032 marksL1 FoundationNon-calculatorTranslate & reflect
      The point (3,−1)(3,-1) is reflected in the line y=xy=x. Find its image.

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      Salam Academy Original Practice · Proposed mark scheme
      SA-4024-TV1-0022 marksL2 StandardNon-calculatorTranslate & reflect
      The point (−3,4)(-3,4) is reflected in the line y=−xy=-x. Find its image.

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      Salam Academy Original Practice · Proposed mark scheme
      SA-4024-TV-0082 marksL2 StandardNon-calculatorTranslate & reflect
      The point (5,1)(5,1) is reflected in the line x=2x=2. Find its image.

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      Salam Academy Original Practice · Proposed mark scheme
      SA-4024-TV-0042 marksL2 StandardNon-calculatorRotations
      The point (4,1)(4,1) is rotated 90∘90^\circ anticlockwise about the origin. Find its image.

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      Salam Academy Original Practice · Proposed mark scheme
      SA-4024-TV1-0032 marksL2 StandardNon-calculatorRotations
      The point (2,3)(2,3) is rotated 90∘90^\circ clockwise about the origin. Find its image.

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      Salam Academy Original Practice · Proposed mark scheme

      More questions on every skill in this subtopic are in the question bank under Practise.

      Section 10 of 14

      Challenge

      SA-4024-TV-0052 marksL3 ExamNon-calculatorRotations
      The point (4,3)(4,3) is rotated 180∘180^\circ about the point (1,2)(1,2). Find its image.

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      Salam Academy Original Practice · Proposed mark scheme
      Section 11 of 14

      Summary

      • Translation: column vector.
      • Reflection: mirror line.
      • Rotation: centre, angle and direction.
      • All three keep size and shape (congruent images).
      Section 12 of 14

      Quick quiz

      Q1 · SA-4024-TV-Q011 markL1 FoundationCalculatorTranslate & reflect
      A translation is fully described by
      Choose one option

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      Salam Academy Original Practice · Proposed mark scheme
      Q2 · SA-4024-TV-Q021 markL1 FoundationCalculatorTranslate & reflect
      Under a reflection, which points do not move?
      Choose one option

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      Salam Academy Original Practice · Proposed mark scheme
      Q3 · SA-4024-TV-0091 markL1 FoundationCalculatorRotations
      To describe a rotation fully, you must give
      Choose one option

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      Salam Academy Original Practice · Proposed mark scheme
      Section 13 of 14

      Same topic, different board

      Specification
      GCSE Mathematics 8300
      Where
      Geometry and measures
      Expected
      The same transformations.
      Calculator
      Paper 1 non-calculator; Papers 2 and 3 calculator.
      Typical style
      Problem-solving questions often combine topics.
      Open in
      8300 · Papers 1–38300 · Papers 1–3
      Section 14 of 14

      Past paper questions

      4 questions (28 marks) from real exams test this subtopic. Try each one under timed conditions, then mark it with the official mark scheme.

      1. 4024/12 · Oct/Nov 2025Question 114 marks

        Describe a transformation and draw a 90° clockwise rotation.

        Question paper, page 8Mark scheme
      2. 4024/12 · May/June 2025Question 128 marks

        Describe and carry out reflections and enlargements on a grid.

        Question paper, page 10Mark scheme
      3. 4024/12 · Oct/Nov 2024Question 165 marks

        Translations and describing transformations.

        Question paper, page 9Mark scheme
      4. 4024/22 · May/June 2024Question 511 marks

        Transformations, enlargement and area scale factor.

        Question paper, page 8Mark scheme

      All 4024 past paper questions by topic

      Salam Academy original content · Content status: reviewed and published. Syllabus mapping: AQA 8300, First teaching 2015 (official specification).