Enlargement: centres and scale factors

An enlargement changes the size of a shape but keeps its angles. The centre and the scale factor tell you exactly where the image goes.

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Scale factor and centre

Scale factor = image length ÷ object length.
From the centre C, each image point is k times as far as the object point, in the same direction.
Centre (1, 1), scale factor 2: P(3, 2) is 2 across and 1 up from C, so the image is 4 across and 2 up: (5, 3).
1 2 3 4 5 6 7 8 1 2 3 4 5 6 x y C (1, 1) P P′ P (3, 2) → P′ (5, 3)object grey, image orange
Enlargement, scale factor 2, centre (1, 1): P(3, 2) goes to (5, 3).

Fractional scale factors

A scale factor between 0 and 1 makes the shape smaller, but it is still called an enlargement.
Scale factor 12: every image point is half as far from the centre.
To find the centre of an enlargement, draw lines through matching corners of the object and image. They cross at the centre.

Negative scale factors (Higher, 1MA1)

A negative scale factor puts the image on the other side of the centre, upside down.
Scale factor −12, centre (0, 0): (4, 6) goes to (−2, −3).
To describe an enlargement fully, give the word 'enlargement', the scale factor and the centre.
−4 −3 −2 −1 1 2 3 4 5 6 7 −4 −3 −2 −1 1 2 3 4 5 6 7 x y (4, 6) (−2, −3) scale factor −½
Negative scale factor −1/2, centre (0, 0): (4, 6) goes to (−2, −3).
Worked example

Enlarge P(3, 2) by scale factor 2 with centre (1, 1).

  1. From the centre to P: +2 across, +1 up
  2. Double it: +4 across, +2 up
  3. Image: (1 + 4, 1 + 2) = (5, 3)

Answer: (5, 3)

Key idea

Image = centre + k × (point − centre). Fractions shrink; negative scale factors flip to the other side. Describe with: enlargement, scale factor, centre.

Key words
Enlargement
A transformation that changes size by a scale factor from a centre of enlargement. Angles stay the same.
Scale factor
The number every length is multiplied by in an enlargement or between similar shapes.

All GCSE and International GCSE Mathematics key terms and definitions →

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Extra practice

2 more questions

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  1. Enlarge (5, 4) by scale factor 3, centre (−1, 2). Find the x-coordinate of the image.

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    Answer: 17

    −1 + 3 × (5 − (−1)) = 17.

  2. Enlarge (4, 2) by scale factor 3, centre (2, −2). Find the y-coordinate of the image.

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    Answer: 10

    −2 + 3 × (2 − (−2)) = 10.

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Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

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