Enlargements and Centres of Enlargement
An enlargement changes size from a fixed centre while preserving angles and shape. For a positive scale factor, corresponding vertices lie on the same ray from the centre. For a negative scale factor they lie on opposite rays. Distances are multiplied by the magnitude of the scale factor.
Foundation & Higher, with a Higher extension. Positive fractional scale factors such as 1/2 are part of the shared Foundation/Higher route. Negative scale factors are the Higher extension. Ratio skills from Understanding and Simplifying Ratios are useful.
Constructing an enlargement from a centre
For a positive scale factor k, draw a ray from the centre through each original vertex. Measure the centre-to-vertex distance and multiply it by k. Negative factors are treated separately below.
If k=2, each image vertex is twice as far from the centre on the same ray. If k=1/2, each image vertex lies halfway from the centre to the original vertex.
An enlargement with k between 0 and 1 is a reduction, but the transformation is still called an enlargement.
Positive fractional scale factors
Suppose centre O is at (0,0), and a vertex is at (8,4). With scale factor 1/2, the image vertex is at (4,2). Distances from the centre are halved.
A fractional scale factor is not automatically Higher. Positive fractions such as 1/2 belong in the Foundation route.
If k=1, every image point stays where it is.
The centre can be anywhere
The centre may lie inside, outside or on the shape. Do not scale coordinates from the origin unless the origin is actually the stated centre.
The reliable method is geometric: centre → original vertex → image vertex all lie on one straight ray for positive k.
Worked example — a centre away from the origin
Enlarge A(5,4) by factor ½ about C(1,2). The movement C→A is (4,2); halve it to (2,1), then add it to C. The image is A′=(1+2,2+1)=(3,3). Scale the movement from the centre, not the coordinates themselves.
Recovering the centre and scale factor
Join each original vertex to its corresponding image vertex and extend the lines. Their common intersection is the centre of enlargement.
This method needs distinct object and image positions. When k=1, they coincide and the image does not determine a unique centre. A vertex at the centre stays fixed for every scale factor.
Then compare corresponding centre-to-vertex distances. If OA=4 cm and OA′=10 cm, the scale factor is 10/4=2.5.
What stays the same — and what changes
Angles stay equal. All corresponding lengths are multiplied by |k|, the magnitude of the scale factor (equal to k when k is positive). Parallel sides remain parallel.
This lesson deals with linear scale factor. The k² and k³ relationships for area and volume belong later in Similar Shapes: Length, Area and Volume.
Higher — negative scale factors
For k<0, the image lies on the opposite ray through the centre. The magnitude |k| still controls the size.
For k=−2, distances double and the image lies on the opposite side of the centre. For k=−1/2, distances halve and reverse direction.
An enlargement with k=−1 places each image point the same distance from the centre on the opposite side. This has the same point-mapping effect as a half-turn about the centre, but in a transformation question you should preserve the enlargement description if that is what is asked.
Geometric enlargement is not a map scale
A map scale compares drawing length with real length. A geometric enlargement is a transformation from a specific centre. The numerical ideas are related through ratio, but the operations are not the same.
For map-scale methods, use Units, Scale Drawings and Maps.
Think Like a Mathematician
The centre is the anchor. If you know the centre and one corresponding vertex pair, the line through those points reveals the direction; the distance ratio reveals the scale factor.
Common Mistakes
Scaling about the origin automatically. Use the stated centre.
Moving every point the same distance. That is translation, not enlargement.
Calling every fractional factor Higher. Positive fractional factors are shared content.
Treating a negative scale factor as a negative length. The sign controls direction through the centre.
Practice — enlargements
1. Centre O is fixed. A vertex is 3 cm from O. Under scale factor 4, how far is its image from O?
3×4=12 cm.
2. A vertex is 10 cm from the centre. Under scale factor 1/2, how far is the image?
5 cm.
3. If k=1, what happens to the shape?
Every point stays in the same position.
4. Centre is the origin. Enlarge (6,−4) by scale factor 1/2.
(3,−2).
5. Centre is (2,1). Point A is (6,3). Enlarge A by scale factor 2.
Vector from centre to A is (4,2). Double to (8,4), then add centre: (10,5).
6. OA=4 and OA′=14. Find the positive scale factor.
14÷4=3.5.
7. How do you locate the centre from a drawn object and image?
Join corresponding vertices and extend the lines. Their common intersection is the centre.
8. Does a positive scale factor 0.4 make a reduction or an enlargement?
It makes a reduction, but the transformation is still called an enlargement.
9. Higher: what does scale factor −2 do to distance and direction?
It doubles the distance and places the image on the opposite ray.
10. Higher: centre O is at the origin. Enlarge (8,2) by scale factor −1/2.
(−4,−1).
11. Why is multiplying every coordinate by 3 wrong if the centre is (5,0)?
That scales about the origin, not about (5,0).
12. What remains equal under enlargement?
Corresponding angles remain equal; lengths are multiplied by the magnitude |k| of the scale factor.
Frequently Asked Questions
Can the centre be inside the shape?
Yes. It may be inside, outside or on the figure.
Is scale factor 1/2 Higher-only?
No. Positive fractional scale factors are part of the shared Foundation/Higher route.
When do area and volume scale factors appear?
Later, in the Higher lesson on similar shapes and k²/k³ relationships.
Continue Learning
Previous: Reflections, Rotations and Translations