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Learning objectives
Core idea
Key formulae
Worked examples
Explore the graph
Common mistakes
Exam tip
Guided practice
Practice questions
Challenge
Summary
Quick quiz
Same topic, different board
Past paper questions

Learn4024Paper 1 · Non-calculatorTransformations and vectorsReflections, rotations and translations

Transformations and vectors · Subtopic 1 of 4

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 $90^\circ$ or $180^\circ$ 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

Each image is congruent to the original. Only reflection reverses it (a mirror image).

Translation by $\begin{pmatrix}a\\b\end{pmatrix}$ .
Reflection in the line … (e.g. $y=x$ , $x=2$ ).
Rotation: angle, direction, centre.

\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 $(3,1)$

Reflect in $y=x$ ; rotate $90^\circ$ clockwise about $O$ ; rotate $180^\circ$ about $O$ .

Goal: Three images.
Strategy: Rules above.

$(1,3)$
Swap.
$(1,-3)$
$(y,-x)$ .
$(-3,-1)$
Both negate.

Answer

(1,3)

(1,-3)

(-3,-1)

✓ CheckAll at distance

\sqrt{10}

from

O

✓.

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.

Join two points to their images.
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.

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

17

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

Reflections

y=x:\ (x,y)\to(y,x)\qquad y=-x:\ (x,y)\to(-y,-x)

Rotations about the origin

90^\circ\text{ anticlockwise}:\ (x,y)\to(-y,x)\qquad180^\circ:\ (x,y)\to(-x,-y)

TrickSwap and sign. For a

90^\circ

rotation about the origin, swap the coordinates, then make one negative. Anticlockwise: negate the new

x

. Clockwise: negate the new

y

(x,y)\to(y,-x)

. Check with

(1,0)

: anticlockwise should send it up to

(0,1)

Section 4 of 14

Worked examples

Example 1 · Reflection in a vertical line0 / 2 steps

Reflect

(5,1)

in the line

x=2

Example 2 · Rotation about another centre0 / 2 steps

Rotate

(4,3)

through

180^\circ

about

(1,2)

Section 5 of 14

Explore the graph

A point and its images

Loading the interactive graph…

Try this

The first point is $(a,b)$ . The others are its reflection in $y=x$ , its $90^\circ$ anticlockwise rotation about the origin, and its $180^\circ$ rotation.
Move $(a,b)$ onto the line $y=x$ . Which image now sits on top of it?
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=2

as if it were the

y

-axis

CorrectMeasure from the line

x=2

The mirror line is not always an axis.

IncorrectConfusing

x=2

and

y=2

Correct

x=2

is vertical

All points with

x

-coordinate

2

IncorrectWriting "turn" or "flip"

Correct"Rotation", "reflection"

Use the correct names.

"It moved"

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

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)

is translated by the vector

\dbinom{4}{-3}

. Find the coordinates of its image.

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)

(-2,6)

. Find the column vector of the translation.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-TV1-0012 marksL1 FoundationNon-calculatorTranslate & reflect

The point

(2,5)

is reflected in the

x

-axis. Find its image.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-TV-0032 marksL1 FoundationNon-calculatorTranslate & reflect

The point

(3,-1)

is reflected in the line

y=x

. Find its image.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-TV1-0022 marksL2 StandardNon-calculatorTranslate & reflect

The point

(-3,4)

is reflected in the line

y=-x

. Find its image.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-TV-0082 marksL2 StandardNon-calculatorTranslate & reflect

The point

(5,1)

is reflected in the line

x=2

. Find its image.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-TV-0042 marksL2 StandardNon-calculatorRotations

The point

(4,1)

is rotated

90^\circ

anticlockwise about the origin. Find its image.

Salam Academy Original Practice · Proposed mark scheme

SA-4024-TV1-0032 marksL2 StandardNon-calculatorRotations

The point

(2,3)

is rotated

90^\circ

clockwise about the origin. Find its image.

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)

is rotated

180^\circ

about the point

(1,2)

. Find its image.

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

Salam Academy Original Practice · Proposed mark scheme

Q2 · SA-4024-TV-Q021 markL1 FoundationCalculatorTranslate & reflect

Under a reflection, which points do not move?

Salam Academy Original Practice · Proposed mark scheme

Q3 · SA-4024-TV-0091 markL1 FoundationCalculatorRotations

To describe a rotation fully, you must give

Salam Academy Original Practice · Proposed mark scheme

Section 13 of 14

Same topic, different board

Specification: 4024 O Level Mathematics
Where: Papers 1 and 2 · 7.1 Transformations
Expected: Recognise, describe and draw reflections (in lines such as $x=a$ , $y=b$ , $y=\pm x$ ), rotations (about any centre, through multiples of $90^\circ$ ) and translations (by a column vector).
Calculator: Paper 1 is non-calculator; Paper 2 allows a calculator.
Typical style: Describe fully the single transformation that maps triangle A onto triangle B.
Open in: 4024 · Paper 1 4024 · Paper 2

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.

4024/12 · Oct/Nov 2025Question 114 marks
Describe a transformation and draw a 90° clockwise rotation.
Question paper, page 8 Mark scheme
4024/12 · May/June 2025Question 128 marks
Describe and carry out reflections and enlargements on a grid.
Question paper, page 10 Mark scheme
4024/12 · Oct/Nov 2024Question 165 marks
Translations and describing transformations.
Question paper, page 9 Mark scheme
4024/22 · May/June 2024Question 511 marks
Transformations, enlargement and area scale factor.
Question paper, page 8 Mark scheme

All 4024 past paper questions by topic

Salam Academy original content · Content status: reviewed and published. Syllabus mapping: Cambridge International 4024, Current syllabus (official specification).