A fractional scale factor makes a shape smaller; a negative scale factor flips it through the centre of enlargement as well as scaling it. Both extend the KS3 idea of enlargement and appear regularly on GCSE Higher papers, in questions that ask you to describe the transformation fully with the scale factor, the centre, and the word "enlargement".
What does a fractional scale factor do?
A scale factor between 0 and 1 (such as 1/2 or 1/3) produces an image that is smaller than the original, but the transformation is still called an enlargement. The shape is not distorted — all angles are preserved and all lengths are multiplied by the same fraction.
For a scale factor of 1/2 about a centre C:
- Each vertex of the image is half the distance from C that the corresponding vertex of the object is.
- All lengths in the image are halved.
- All areas in the image are multiplied by (1/2)² = 1/4.
How do you enlarge a shape with a fractional scale factor step by step?
Worked example: Enlarge triangle OAB with vertices O(0, 0), A(4, 0), B(0, 6) by scale factor 1/2 about the centre of enlargement C(0, 0).
- Since C is the origin, multiply each coordinate by 1/2:
- O′ = (0 × 1/2, 0 × 1/2) = (0, 0)
- A′ = (4 × 1/2, 0 × 1/2) = (2, 0)
- B′ = (0 × 1/2, 6 × 1/2) = (0, 3)
- Draw the image triangle O′A′B′.
- Check: each side of the image is half the length of the original side. OA = 4, O′A′ = 2 ✓. OB = 6, O′B′ = 3 ✓.
When the centre is not the origin:
- Find the vector from C to each vertex of the object.
- Multiply that vector by the scale factor.
- Add the result to C to find each image vertex.
For vertex A(6, 4) with C(2, 2) and scale factor 1/2:
- Vector CA = (6−2, 4−2) = (4, 2).
- Multiply by 1/2: (2, 1).
- Image: A′ = C + (2, 1) = (2+2, 2+1) = (4, 3).
What does a negative scale factor do?
A negative scale factor still multiplies all lengths by its absolute value, but it also rotates the image 180° about the centre of enlargement. The result is an image on the opposite side of C from the object, often described as being "through" the centre.
Key differences between positive and negative scale factors:
| Feature | Positive scale factor (e.g. 2) | Negative scale factor (e.g. −2) |
|---|---|---|
| Image size | Lengths × | k |
| Image position | Same side of C as object | Opposite side of C |
| Image orientation | Same way up | Upside down (rotated 180°) |
| Still called | Enlargement | Enlargement |
How do you perform an enlargement with a negative scale factor?
Worked example: Enlarge point P(5, 3) by scale factor −2 about centre C(1, 1).
- Find the vector from C to P: CP = (5−1, 3−1) = (4, 2).
- Multiply by −2: (−8, −4).
- Image: P′ = C + (−8, −4) = (1−8, 1−4) = (−7, −3).
The image ends up on the opposite side of C, and the distance CP′ = 8√2 is twice CP = 4√2. ✓
Worked example for a shape: Triangle with vertices A(3, 4), B(5, 4), C(5, 6), enlarged by scale factor −1/2 about centre O(1, 2).
| Vertex | Vector from O | × (−1/2) | Image |
|---|---|---|---|
| A(3, 4) | (2, 2) | (−1, −1) | A′(0, 1) |
| B(5, 4) | (4, 2) | (−2, −1) | B′(−1, 1) |
| C(5, 6) | (4, 4) | (−2, −2) | C′(−1, 0) |
How do you describe an enlargement fully?
A complete description of an enlargement must include three elements:
- The transformation type: enlargement.
- The scale factor (with sign if negative).
- The centre of enlargement as a coordinate.
Missing any element loses a mark. "It has been enlarged by −2" without stating the centre is an incomplete answer. The centre and scale factor together fully define where every image point goes.
How do you find the centre of enlargement from a diagram?
- Draw lines from each vertex of the image back to the corresponding vertex of the object.
- Extend these lines — they all pass through the same point.
- That point is the centre of enlargement.
If the scale factor is negative, the lines pass through C and continue to the other side. The object and image are on opposite sides of C, so extend the lines in both directions until they meet.
Frequently asked questions
Is a scale factor of −1 the same as a rotation?
Yes — an enlargement with scale factor −1 about a centre C is identical to a rotation of 180° about C. The image is the same size as the object (no change in lengths), but every point is reflected through C. At GCSE, such a transformation can validly be described as either a rotation of 180° about C or an enlargement with scale factor −1 about C.
What happens to the area when a fractional scale factor is used?
If the linear scale factor is k, the area scale factor is k². For a scale factor of 1/3, every length is multiplied by 1/3 and every area is multiplied by (1/3)² = 1/9. This is the same rule as for positive integer scale factors — the square of the linear scale factor always gives the area scale factor.
Can the scale factor be a negative fraction, such as −1/2?
Yes. A scale factor of −1/2 reduces lengths by half (like a scale factor of 1/2) and also rotates the image 180° through the centre (like a negative factor). The absolute value |−1/2| = 1/2 governs the size; the negative sign governs the orientation. All four cases — positive integer, positive fraction, negative integer, negative fraction — work by the same rule.
How do I know whether to multiply or use vectors?
When the centre of enlargement is the origin (0, 0), multiplying every coordinate by the scale factor is a shortcut. When the centre is anywhere else, the vector method (find the vector from C to each vertex, scale it, add back to C) is more reliable and works in all cases.
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