Enlargement and Centre of Enlargement –

Enlargement is the transformation that changes the size of a shape while keeping it the same proportions. It is the only transformation that does not preserve congruence. Enlargement questions appear on both Foundation and Higher GCSE Maths papers, with fractional and negative scale factors reserved for Higher. This guide covers integer, fractional, and negative scale factors, finding the centre of enlargement, and describing enlargements fully.

What Is an Enlargement?

An enlargement scales a shape from a fixed point called the centre of enlargement. Every point on the image is a fixed multiple (the scale factor) of its distance from the centre compared to the original shape.

Key Formulas

A scale factor greater than 1 makes the shape larger. A scale factor between 0 and 1 makes it smaller. A negative scale factor (Higher) produces an image on the opposite side of the centre and inverted.

Step-by-Step Method

1. Identify the centre of enlargement and the scale factor from the question.
2. For each vertex, draw a ray from the centre through the vertex.
3. Measure the distance from the centre to the vertex and multiply by the scale factor.
4. Plot the image vertex at the new distance along the same ray (or the opposite direction for negative scale factors).
5. Join the image vertices to complete the enlarged shape.

Worked Example 1 — Foundation Level

Question: Enlarge the triangle with vertices A(1, 1), B(3, 1), and C(1, 3) by scale factor 2 from the centre (0, 0).

Working:

Multiply each coordinate by the scale factor.

A(1, 1) maps to A'(2, 2)

B(3, 1) maps to B'(6, 2)

C(1, 3) maps to C'(2, 6)

Answer: The image has vertices A'(2, 2), B'(6, 2), C'(2, 6).

Worked Example 2 — Higher Level

Question: Enlarge the shape with vertex P(4, 6) by scale factor 1/2 from centre (2, 2).

Working:

Vector from centre to P: (4 − 2, 6 − 2) = (2, 4)

Multiply by 1/2: (1, 2)

Image P' = (2 + 1, 2 + 2) = (3, 4)

Answer: P' = (3, 4).

Worked Example 3 — Exam Style

Question: Shape A is enlarged to give shape B. A vertex of A is at (1, 2) and the corresponding vertex of B is at (−3, 6). The centre of enlargement is (3, 0). Find the scale factor.

Working:

Vector from centre to A vertex: (1 − 3, 2 − 0) = (−2, 2)

Vector from centre to B vertex: (−3 − 3, 6 − 0) = (−6, 6)

Scale factor = −6 ÷ −2 = 3 (check: 6 ÷ 2 = 3)

Answer: The scale factor is 3.

Common Mistakes

• Forgetting to use the centre of enlargement. Multiplying the coordinates by the scale factor only works when the centre is (0, 0). For any other centre, use vectors from the centre first.
• Drawing rays in the wrong direction for negative scale factors. A negative scale factor means the image is on the opposite side of the centre. The ray goes through the centre and out the other side.
• Incomplete description. To describe an enlargement fully, state the transformation type (enlargement), the scale factor, and the centre. All three are required.

Exam Tips

• When finding the centre of enlargement, draw rays from corresponding vertices of the object and image back until they meet. The intersection is the centre.
• A scale factor of −1 produces a shape the same size on the opposite side of the centre — this is equivalent to a 180° rotation about that point.
• Check your answer by verifying that corresponding sides of the image are in the correct ratio to the original.
• On Higher papers, expect questions combining fractional or negative scale factors with finding the centre.

Practice Questions

Q1 (Foundation): Enlarge the point (2, 3) by scale factor 3 from the origin.

Q2 (Foundation): A shape is enlarged so that a side of 4 cm on the object becomes 10 cm on the image. What is the scale factor?

Q3 (Higher): Enlarge the point (5, 3) by scale factor −2 from the centre (1, 1). Find the image coordinates.

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Related Topics

• Similar Shapes and Scale Factors
• Reflections
• Rotations
• Translations and Column Vectors

Summary

• An enlargement scales a shape from a centre of enlargement by a given scale factor. Scale factor greater than 1 makes the shape larger, between 0 and 1 makes it smaller, and a negative scale factor inverts the image through the centre. To describe an enlargement fully, state the type, scale factor, and centre. Always work with vectors from the centre to each vertex, then multiply by the scale factor to find image positions.