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GCSE Maths · Geometry & measures

Enlargement GCSE Questions and Worked Answers

An enlargement scales every distance from a fixed centre by the same factor. Draw a ray from the centre through each vertex and place its image at the scaled distance. A complete description gives both centre and scale factor.

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Start with the meaning

What you need to know about enlargement

Imagine a shape projected from one fixed point. To make an image twice as large, move each corner twice as far from that point along the same direction. The fixed point is the centre of enlargement. The multiplier, here 2, is the scale factor. Corners are also called vertices.

See the idea first

Scale the journey from the centre

In the diagram, O = (0, 0) is the centre and the factor is 2. The corner at (1, 2) becomes (2, 4): its movement from O doubles in both directions. The same multiplier must be used for every corner. The shape keeps its angles and proportions.

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Every vertex moves twice as far from O along its own ray. Matching side lengths also double. Grid squares have equal horizontal and vertical units.
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.

Draw a positive enlargement

What the problem asks: Enlarge P = (3, 2) by factor 2 from C = (1, 1).

How to solve it: From C to P, move 2 right and 1 up. Double that journey to 4 right and 2 up, starting at C: P′ = (5, 3).

Use a fractional factor

What the problem asks: Enlarge P = (6, 4) by factor 1/2 from the origin.

How to solve it: Take half of each movement from the origin, giving P′ = (3, 2). An enlargement can make a shape smaller.

Describe an enlargement

What the problem asks: An image side is 9 cm and its matching original side is 3 cm. What else is needed?

How to solve it: The size ratio has magnitude 9/3 = 3. Use the positions to decide the sign, and extend lines through matching vertices to locate the common centre. Side lengths alone do not identify the centre.

Use a negative factor — Higher

What the problem asks: Enlarge P = (2, 1) by factor −2 from the origin.

How to solve it: Double the distance and reverse the direction from the centre. P′ = (−4, −2). The image is on the opposite side of the centre.

A reliable routine

Enlarge a shape from a given centre

Use this when the centre and scale factor are known. Scaling each centre-to-vertex journey by the same factor preserves angles and scales every side consistently.

  1. Mark the centre and choose a vertex.
  2. Find its horizontal and vertical displacement from the centre.
  3. Multiply both displacements by the scale factor, then start that new journey at the centre.
  4. Repeat for every vertex and join images in the same order.

Check: If the centre is not (0, 0), multiplying the vertex coordinates directly gives the wrong image. A prime mark, P′, simply labels the image of P.

Fully worked

Enlargement GCSE worked examples

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

Example 1

Origin as centre

2 marks
Question

Enlarge P = (2, 3) by factor 3 from (0, 0).

The displacement is (2, 3). Triple it:

P′=(3⋅2, 3⋅3)=(6,9)P'=(3\cdot2,\ 3\cdot3)=(6,9)
Example 2

A different centre

3 marks
Question

Enlarge P = (5, 4) by factor 2 from C = (2, 1).

Subtract the centre:

P−C=(3,3)P-C=(3,3)

Double the displacement: (6, 6). Add it back to C:

P′=(2+6,1+6)=(8,7)P'=(2+6,1+6)=(8,7)
Example 3

A whole triangle

3 marks
Question

Enlarge triangle A(1, 1), B(3, 1), C(1, 2) by factor 2 from the origin.

Double every coordinate because the centre is the origin.

A′=(2,2),B′=(6,2),C′=(2,4)A'=(2,2),\quad B'=(6,2),\quad C'=(2,4)

Join A′B′C′. The original base length 2 becomes 4; the vertical side 1 becomes 2.

Example 4

Fractional scale factor

3 marks
Question

Enlarge P = (7, 5) by factor 1/2 from C = (1, 1).

The displacement from C is (6, 4). Half is (3, 2).

P′=(1+3,1+2)=(4,3)P'=(1+3,1+2)=(4,3)

The image is halfway between C and P.

Example 5

Negative factor

Higher only3 marks
Question

Enlarge P = (4, 2) by factor −2 from C = (1, 1).

Displacement: (3, 1). Multiply by −2: (−6, −2).

P′=(1−6,1−2)=(−5,−1)P'=(1-6,1-2)=(-5,-1)

The image lies on the other side of C, twice as far away.

Example 6

Find the centre

4 marks
Question

An enlargement has factor 2. It maps P = (3, 2) to P′ = (5, 3). Find the centre C.

For factor 2, the centre-to-P displacement equals the P-to-P′ displacement.

P′−P=(2,1)P'-P=(2,1)

Travel that amount backwards from P:

C=(3−2,2−1)=(1,1)C=(3-2,2-1)=(1,1)

Check: from C, (2, 1) doubles to (4, 2), reaching (5, 3).

10 original questions · total 25 marks

Enlargement GCSE exam-style questions

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

Before you startAllow about 30 minutes · Show your working. Give exact answers unless a question asks you to round, and include units where needed. · answers start collapsed
1

Factor 4

2 marks

Enlarge (1, 2) by factor 4 from the origin.

Show worked answer
P′=(4⋅1,4⋅2)=(4,8)P'=(4\cdot1,4\cdot2)=(4,8)
2

Negative coordinate

2 marks

Enlarge (−2, 3) by factor 2 from the origin.

Show worked answer
P′=(−4,6)P'=(-4,6)

The direction from the centre stays the same because the factor is positive.

3

Half size

2 marks

Enlarge (8, −6) by factor 1/2 from the origin.

Show worked answer
P′=(4,−3)P'=(4,-3)

Halve both displacements.

4

Non-origin centre

3 marks

Enlarge (4, 5) by factor 3 from (1, 2).

Show worked answer

Displacement (3, 3) becomes (9, 9).

P′=(1+9,2+9)=(10,11)P'=(1+9,2+9)=(10,11)
5

Fraction from a centre

3 marks

Enlarge (10, 6) by factor 1/3 from (1, 0).

Show worked answer

Displacement (9, 6) becomes (3, 2).

P′=(1+3,0+2)=(4,2)P'=(1+3,0+2)=(4,2)
6

Side lengths

2 marks

An enlargement with positive factor maps a 4 cm side to 10 cm. Find the factor.

Show worked answer
k=104=2.5k=\frac{10}{4}=2.5

Divide image length by original length.

7

A fixed point

2 marks

What happens to the centre itself under an enlargement of factor 5?

Show worked answer

It stays fixed. Its displacement from itself is (0, 0), and multiplying both zeros by 5 still gives (0, 0).

8

Negative half

Higher only3 marks

Enlarge (6, −4) by factor −1/2 from the origin.

Show worked answer
P′=(−12(6),−12(−4))=(−3,2)P'=\left(-\frac12(6),-\frac12(-4)\right)=(-3,2)
9

Negative with a centre

Higher only3 marks

Enlarge (3, 4) by factor −1 from (1, 1).

Show worked answer

Displacement (2, 3) reverses to (−2, −3).

P′=(1−2,1−3)=(−1,−2)P'=(1-2,1-3)=(-1,-2)
10

Complete a description

3 marks

A triangle is enlarged from (2, 1), with all image vertices on the same rays and three times as far from the centre. Describe the transformation fully.

Show worked answer

Enlargement, scale factor 3, centre (2, 1). The same-ray condition makes the factor positive; the distance ratio supplies its magnitude.

Examiner-style feedback

Common enlargement mistakes

Ignoring the centre

Scale displacements from the given centre, not coordinates measured from the origin.

Adding the factor

A factor multiplies distances; it does not add a fixed number to every coordinate.

Missing sign or centre

Give both the signed factor and the centre when describing an enlargement.

30-second recap

Choose, carry out, check

Translate the wording before you reach for a rule.

  1. Centre first.
  2. Scale centre-to-vertex journeys.
  3. Repeat for all vertices.
  4. Positive factors between 0 and 1 shrink; negative factors reverse direction.
Quick answers

Enlargement FAQ

Can an enlargement make a shape smaller?

Yes. A positive factor between 0 and 1 moves vertices closer to the centre.

Does factor 2 double the area?

No. It doubles each length, so area is multiplied by 2² = 4.

Build connected skills

What to revise next

Browse all GCSE Maths study guidesReview the worked examplesTry the questions again
Content standards

Curriculum and rights review

G7 enlargements across tiers. Negative scale factors are Higher-only and labelled on questions. Questions, suggested marks, solutions and diagrams are original practice material, not official past-paper questions.

Official specification references