Transformations GCSE Questions and Worked Answers

Free AI tutor · GCSE transformations

Practise GCSE transformations for free with an AI tutor

Ask Ari for an explanation or work through an original exam-style question together.

Hi, I’m Ari. We can move one corner on a grid, then use that idea to translate, reflect, rotate or enlarge a whole shape.

Ari is an AI tutor and can make mistakes. Use the worked answers below to check important results.

Begin with a point on a grid

What you need to know about transformations

Imagine a triangle drawn on squared paper. Move one corner from (1, 1) to (3, 3): it travels 2 squares right and 2 up. Coordinates give a point’s horizontal position first and vertical position second, measured from (0, 0), the origin. If every corner makes this same move, the whole triangle slides. This particular transformation is called a translation.

Follow one corner, then every corner

The original shape and its image

A corner is also called a vertex. The shape after a transformation is its image. A′ (‘A prime’) labels the image of A, so the matching corners remain easy to follow.

Original cornerA = (1, 1)first across, then up

Move2 right, 2 upmake the same move at B and C

Image cornerA′ = (3, 3)join the new corners in the same order

Translation: every corner moves 2 right and 2 up. Dashed ABC is the original; solid A′B′C′ is its image.

From problem to method

Typical problems you need to be able to solve

These are the main kinds of question in this topic. Each card first states the problem, then explains how to solve it.

Translate a shape

What the problem asks: Move triangle ABC 2 units right and 2 units up.

How to solve it: Add 2 to each x-coordinate and 2 to each y-coordinate. Every point has the same displacement, so lengths and angles are preserved.

\begin{pmatrix}2\\2\end{pmatrix}\text{ means 2 right, 2 up.}

Reflect a shape

What the problem asks: Draw the mirror image of a triangle in the y-axis.

How to solve it: For each corner, count the shortest distance to the mirror line. Place its image equally far on the other side, along a line perpendicular (at 90°) to the mirror. Points on the mirror stay fixed.

Rotate a shape

What the problem asks: Turn a triangle 90° anticlockwise about (0, 0).

How to solve it: Keep the centre fixed. Turn every corner through the same angle in the same direction, keeping its distance from the centre. Tracing paper pinned at the centre helps.

Enlarge a shape

What the problem asks: Enlarge triangle ABC by scale factor 2 about (0, 0).

How to solve it: Draw a ray from the centre through each vertex. Double each distance from the centre. A scale factor of 1/2 halves the distances; a negative scale factor places the image on the opposite side (Higher).

Describe one transformation completely

What the problem asks: Explain exactly how triangle ABC becomes A′B′C′.

How to solve it: Give a translation vector; a reflection’s mirror line; a rotation’s centre, angle and direction; or an enlargement’s centre and scale factor. Verify the description using all corners.

A reliable routine

Construct an enlargement from its centre

This method is for an enlargement with a given centre and scale factor. It works by multiplying every displacement from the centre by the same factor, keeping corresponding angles equal and scaling side lengths uniformly. The other three transformations use their own rules above.

Mark the given centre; it is not necessarily the origin or a corner of the shape.
Count the horizontal and vertical steps from the centre to one vertex.
Multiply both step counts by the scale factor.
Starting at the same centre, follow those new steps to mark the image vertex.
Repeat for every vertex and join the image vertices in the original order.

Check: For positive scale factors between 0 and 1 the image is smaller. ‘Enlargement’ is still the mathematical name. Negative factors and combinations of transformations are Higher extensions.

Fully worked

Transformations GCSE worked examples

Each solution explains the clue, why the method fits and how to check the result.

Example 1

Translate a triangle

3 marks

Question

A(1,1), B(3,1), C(1,2) are translated by the vector

\begin{pmatrix}2\\2\end{pmatrix}

Find the image coordinates.

The top vector entry changes x; the bottom changes y. Add 2 to both coordinates of each point.

A'=(1+2,1+2)=(3,3)

B'=(3+2,1+2)=(5,3)

C'=(1+2,2+2)=(3,4)

Check that each corner made the same move.

Translation: every corner moves 2 right and 2 up. Dashed ABC is the original; solid A′B′C′ is its image.

Example 2

Reflect in the y-axis

3 marks

Question

Reflect A(1,1), B(3,1), C(1,2) in the y-axis. Give the image coordinates.

The y-axis is the vertical line $x=0$ . Reflecting across it reverses horizontal position but keeps height unchanged.

A' = (-1,1)

B' = (-3,1)

C' = (-1,2)

Each original and image corner is the same distance from $x=0$ .

Reflection in the y-axis: matching corners sit at equal perpendicular distances from the mirror line, on opposite sides.

Example 3

Rotate a quarter turn

3 marks

Question

Rotate A(1,1), B(3,1), C(1,2) through $90^\circ$ anticlockwise about the origin.

About the origin, a move right turns into a move up; a move up turns into a move left. Thus $(x,y)$ becomes $(-y,x)$ .

A'=(-1,1)

B'=(-1,3)

C'=(-2,1)

Tip: that coordinate shortcut is for this angle and centre, not every rotation.

Rotation 90° anticlockwise about (0, 0): each corner turns a quarter turn and keeps the same distance from the centre.

Example 4

Enlarge from the origin

3 marks

Question

Enlarge A(1,1), B(3,1), C(1,2) by scale factor 2, centre (0,0).

The coordinates already measure displacement from the centre, so double both coordinates.

A'=(2,2)

B'=(6,2)

C'=(2,4)

AB had length 2 and A′B′ has length 4: the scale factor doubles lengths.

Enlargement, scale factor 2, centre (0, 0): each image corner is twice as far from the centre in the same direction.

Example 5

Use a centre away from the origin

3 marks

Question

Point P(3,2) is enlarged by scale factor 2 about centre (1,1). Find P′.

Measure from the centre first.

\text{displacement}=(3-1,2-1)=(2,1)

Double that displacement: $(4,2)$ . Start again at the centre $(1,1)$ .

P'=(1+4,1+2)

\boxed{P'=(5,3)}

Doubling P’s original coordinates would incorrectly give (6,4).

Example 6

Use a negative scale factor

Higher only3 marks

Question

Enlarge A(1,1), B(3,1), C(1,2) by scale factor −1, centre (0,0).

Multiply each displacement from the centre by $-1$ . This reverses its direction without changing its length.

A'=(-1,-1)

B'=(-3,-1)

C'=(-1,-2)

Here the result is also a 180° rotation about the origin; it is not a reflection in either axis.

Enlargement, scale factor −1, centre (0, 0): each image corner is equally far from the centre on the opposite side. This Higher example has the same size as the original.

10 original questions · total 26 marks

Transformations GCSE exam-style questions

Try each question before opening its fully worked answer. The difficulty rises through the set.

Before you startAllow about 35 minutes · Use squared paper. Label original and image points, show the centre or mirror line, and give all information needed when describing a transformation. · answers start collapsed

1

Apply a translation vector

2 marks

Translate P(−2,4) by $\begin{pmatrix}5\\-3\end{pmatrix}$ .

Show worked answer

Move 5 right and 3 down.

P'=(-2+5,4-3)

\boxed{P'=(3,1)}

2

Describe a translation

2 marks

Every corner of a shape makes the same move. A(−1,2) moves to A′(3,−1). Describe the translation.

Show worked answer

Subtract old coordinates from new coordinates.

\Delta x=3-(-1)=4

\Delta y=-1-2=-3

Translation by $\boxed{\begin{pmatrix}4\\-3\end{pmatrix}}$ .

3

Reflect in a horizontal line

2 marks

Reflect P(2,5) in the line y = 1.

Show worked answer

P is $5-1=4$ units above the mirror. The image is 4 units below it, with the same x-coordinate.

y'=1-4=-3

\boxed{P'=(2,-3)}

4

Reflect in a diagonal line

2 marks

Reflect P(−2,4) in the line y = x.

Show worked answer

The diagonal mirror exchanges horizontal and vertical positions: $(x,y)$ becomes $(y,x)$ .

\boxed{P'=(4,-2)}

The midpoint (1,1) lies on y = x, and the segment PP′ is perpendicular to the mirror.

5

Rotate clockwise

2 marks

Rotate P(2,−3) by 90° clockwise about (0,0).

Show worked answer

A clockwise quarter turn maps $(x,y)$ to $(y,-x)$ .

\boxed{P'=(-3,-2)}

The point stays the same distance from the origin.

6

Use a fractional scale factor

3 marks

Enlarge P(6,4) by scale factor 1/2 about (2,0).

Show worked answer

From the centre to P, the displacement is $(4,4)$ . Half of this is $(2,2)$ .

P'=(2+2,0+2)

\boxed{P'=(4,2)}

The image lies halfway from the centre to P.

7

Find the centre of enlargement

Harder4 marks

An enlargement with scale factor 2 maps P(3,2) to P′(4,3), and Q(4,2) to Q′(6,3). Find its centre.

Show worked answer

For factor 2, the move from the centre to P equals the move from P to P′. P to P′ is $(1,1)$ , so step back $(1,1)$ from P.

\text{centre}=(3-1,2-1)=(2,1)

Check Q: from the centre $(2,1)$ to Q is the displacement $(2,1)$ . Doubling gives the new displacement $(4,2)$ , not the image coordinates. Start at the centre and add this displacement:

Q'=(2+4,1+2)=(6,3)

\boxed{\text{centre }(2,1)}

On a grid, extend PP′ and QQ′ backwards; they meet at the centre.

8

Describe the illustrated reflection

2 marks

Describe the single reflection mapping dashed ABC to solid A′B′C′.

Reflection in the y-axis: matching corners sit at equal perpendicular distances from the mirror line, on opposite sides.

Show worked answer

Corresponding points are equally far to either side of the vertical line $x=0$ .

Reflection in the y-axis, or reflection in the line $x=0$ . Include the mirror line, not just ‘reflection’.

9

Enlarge with a negative factor

Higher only3 marks

Enlarge P(3,2) by scale factor −2 about (1,1).

Show worked answer

The displacement from the centre is $(2,1)$ . Multiplying it by $-2$ gives $(-4,-2)$ .

P'=(1-4,1-2)

\boxed{P'=(-3,-1)}

The image is on the opposite side of the centre, at twice the distance.

10

Carry out two transformations in order

Higher only4 marks

P(1,3) is translated 2 right, then reflected in the y-axis. Find the final point. Would reversing the order give the same result?

Show worked answer

Translation first:

(1,3)\to(3,3)\to(-3,3)

Reflection first:

(1,3)\to(-1,3)\to(1,3)

The requested final point is $\boxed{(-3,3)}$ . Reversing the order gives a different point. A second transformation acts on the first image, not the original.

Examiner-style feedback

Common transformations mistakes

Giving an incomplete description

‘Rotation’ needs a centre, angle and direction. ‘Enlargement’ needs a centre and scale factor.

Multiplying coordinates about the wrong centre

Only multiply coordinates directly if the centre is the origin. Otherwise measure from the centre, scale that displacement, then add it back.

Treating a negative factor as a reflection

A negative enlargement sends each point through the centre. An axis reflection flips only one coordinate about that axis.

Forgetting what stays unchanged

Translations, reflections and rotations preserve lengths and angles. Enlargements preserve angles; lengths scale by the magnitude of the scale factor.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

Match original vertices to their images.
Use the rule for the specified transformation.
State the centre, mirror line or vector precisely.
Check every corner, not just one.

Quick answers

Transformations FAQ

Which transformations are Higher only?

Negative scale factors and the effects of combinations of rotations, reflections and translations are Higher content. The four single transformations, including positive fractional enlargements, occur across tiers.

Can an enlargement make a shape smaller?

Yes. A positive scale factor between 0 and 1 shortens every displacement from the centre.

What is an invariant point?

It is a point that stays in exactly the same place. For example, a point on a reflection’s mirror line is invariant.

What must a rotation description include?

State its centre, angle and direction. For 180° the clockwise and anticlockwise results are the same.

Build connected skills

What to revise next

Translation notation

Vectors

Read and combine horizontal and vertical displacements.

Revise vectors

Scaling

Ratio

Connect scale factors to proportional lengths.

Revise ratio

Course map

GCSE Maths topics

Choose your next geometry skill.

Revise gcse maths topics

Content standards

Curriculum and rights review

Reviewed 7 September 2026 against GCSE G7, G8 and G24. All coordinates, diagrams and questions are original. Higher variants are labelled locally; suggested marks are Pass an Exam estimates.

Official specification references

AQA GCSE geometry: transformations and vectors