Enlargement with Negative Scale Factor

What a Negative Scale Factor Does

Most students are comfortable with positive enlargements — the image gets bigger (scale factor greater than 1) or smaller (scale factor between 0 and 1), and it stays on the same side of the centre of enlargement. A negative scale factor adds a twist: the image appears on the opposite side of the centre, and it is also inverted (rotated 180°).

Think of it like this: a negative scale factor combines an enlargement with a rotation of 180° about the centre. This is what makes these questions tricky, and it is why they appear frequently on IGCSE Extended Paper 4.

The Method

To enlarge a shape by a negative scale factor k from a centre of enlargement C:

For each vertex of the original shape, draw a line from C through the vertex
Measure the distance from C to the vertex
Multiply this distance by |k| (the absolute value of the scale factor)
Plot the image point on the opposite side of C from the original vertex, at the new distance

Alternatively, you can use a coordinate method:

Image point = Centre + k × (Original point − Centre)

This formula works for both positive and negative scale factors and is more reliable in an exam.

Worked Example

Triangle PQR has vertices P(2, 1), Q(4, 1), and R(3, 3). Enlarge the triangle by scale factor −2 with centre of enlargement (1, 2).

Step 1: Find the vector from the centre to each vertex.

Centre C = (1, 2).

CP = P − C = (2 − 1, 1 − 2) = (1, −1)
CQ = Q − C = (4 − 1, 1 − 2) = (3, −1)
CR = R − C = (3 − 1, 3 − 2) = (2, 1)

Step 2: Multiply each vector by the scale factor (−2).

−2 × CP = −2 × (1, −1) = (−2, 2)
−2 × CQ = −2 × (3, −1) = (−6, 2)
−2 × CR = −2 × (2, 1) = (−4, −2)

Step 3: Add these new vectors to the centre to find the image points.

P’ = C + (−2, 2) = (1 + (−2), 2 + 2) = (−1, 4)
Q’ = C + (−6, 2) = (1 + (−6), 2 + 2) = (−5, 4)
R’ = C + (−4, −2) = (1 + (−4), 2 + (−2)) = (−3, 0)

Step 4: Draw and check.

The image triangle P’Q’R’ has vertices at (−1, 4), (−5, 4), and (−3, 0). Notice:

The image is on the opposite side of the centre from the original — the original was mainly to the right of C, while the image is to the left
The image is twice the size of the original (|scale factor| = 2)
The image is inverted — it appears upside-down relative to the original
The centre of enlargement (1, 2) lies on the line connecting each original vertex to its image

Describing an Enlargement

If the exam gives you the original shape and its image, you may need to describe the transformation. To identify that it is an enlargement with a negative scale factor:

Check if the image is inverted. If the shape appears to have been rotated 180° as well as enlarged, the scale factor is negative.
Find the centre. Draw lines connecting each vertex to its image. The centre of enlargement is where these lines intersect.
Find the scale factor. Measure the distance from the centre to an image point and divide by the distance from the centre to the corresponding original point. Check the sign — if the image is on the opposite side, the scale factor is negative.

Example of finding the scale factor:

Distance from C(1, 2) to P(2, 1) = √(1² + 1²) = √2

Distance from C(1, 2) to P’(−1, 4) = √(2² + 2²) = √8 = 2√2

Scale factor = −(2√2 / √2) = −2 (negative because P’ is on the opposite side of C from P).

What the Examiner Expects

When describing an enlargement, you must state three things:

The type of transformation: enlargement
The scale factor: including the sign (e.g., −2)
The centre of enlargement: as coordinates (e.g., (1, 2))

Missing any one of these costs you marks. In particular, forgetting the negative sign or not stating the centre are very common errors.

Special Cases

Scale factor −1: The image is the same size as the original but on the opposite side of the centre. This is equivalent to a rotation of 180° about the centre.
Scale factor between −1 and 0 (e.g., −0.5): The image is smaller than the original, inverted, and on the opposite side of the centre.
Scale factor 0: This maps every point to the centre. This is not useful and will not appear in an exam.

Common Mistakes

Plotting the image on the same side as the original. A negative scale factor always puts the image on the opposite side of the centre.
Forgetting the negative sign in the description. If you write “enlargement, scale factor 2, centre (1, 2),” you lose marks because the sign matters.
Measuring distances incorrectly. Use the vector method — it is more reliable than measuring on graph paper, especially when the scale factor is not a whole number.
Not identifying the centre correctly. When describing a given enlargement, draw at least two lines connecting original vertices to image vertices. The centre is their intersection.
Confusing enlargement with rotation. A negative enlargement looks like it includes a rotation, but it is classified as an enlargement, not a combined transformation.

Practice Question

A triangle has vertices at A(1, 1), B(3, 1), and C(2, 4). Enlarge the triangle by scale factor −1.5 with centre of enlargement (0, 0).

Find the coordinates of A’, B’, and C’. Then describe two other single transformations that would map triangle ABC to triangle A’B’C’ if the scale factor were exactly −1.

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