The Enlargement Transformation
Resizing a shape around a centre
An enlargement is a transformation that resizes a shape, making it larger or smaller while keeping it recognisably the same shape. It is governed by two ingredients: a fixed point called the centre of enlargement, and a number called the scale factor. Every point of the original moves directly away from, or towards, the centre, and the scale factor tells you by how much. The result is a new shape, the image, that is similar to the original, which links this unit directly to the idea of similarity met earlier.
The enlargement rule
The rule itself is precise. Take the centre of enlargement and a point of the shape, and draw the line joining them. The image of that point lies on the same line, at a distance from the centre equal to the scale factor times the original distance. With a centre at the origin and a scale factor of two, a point at one comma one maps to two comma two, and a point at three comma one maps to six comma two; each image is simply twice as far from the centre along the same ray. Dynamic geometry software makes this vivid: drag the scale factor and watch the whole shape breathe in and out around the fixed centre.
The scale factor: enlarge, reduce, or keep
The scale factor controls the size of the change. A factor greater than one produces a genuine enlargement, so a factor of two doubles every distance from the centre. A factor between zero and one produces a reduction, shrinking the shape, so a factor of one half halves every distance, sending eight comma two to four comma one. A scale factor of exactly one leaves the shape unchanged, since every point stays where it is. Choosing the factor is choosing how dramatic the resize will be.
How lengths change
The heart of this topic is sorting out what changes and what does not. Lengths change in the most direct way: every side of the image is the scale factor times the matching side of the original. Under a scale factor of three, a five centimetre side becomes fifteen centimetres, and the perimeter, being a sum of lengths, also multiplies by three. So far the pattern is simple multiplication by the scale factor.
Why area scales by the factor squared
Area behaves differently, and this is the most common trap. When lengths multiply by the scale factor, area multiplies by the scale factor squared, because area depends on two dimensions at once. A shape enlarged by a factor of two has sides twice as long but an area four times as large, not twice. A triangle of area six becomes an image of area twenty-four under a factor of two. Forgetting to square the scale factor for area is the single most frequent mistake, so it is worth stating plainly: lengths use the factor, areas use the factor squared.
What stays the same: the invariants
Against these changes stand the quantities that stay exactly the same, the invariants. The angles of the shape are unchanged, which is precisely why the image is similar to the original rather than distorted. The overall shape is preserved, parallel sides remain parallel, and the ratios between lengths within the figure are untouched. An enlargement can move a shape and resize it, but it can never bend an angle or alter the proportions inside it. This is the formal reason an enlargement always produces a similar figure.
A checklist for any enlargement
Putting it together gives a clear checklist for any enlargement problem. Identify the centre and the scale factor first. To find an image point, move along the ray from the centre by the scale factor. To find a new length or perimeter, multiply by the scale factor; to find a new area, multiply by the scale factor squared. Then name the invariants: angles, shape, parallelism and internal ratios stay fixed. Holding the changing quantities and the unchanging ones clearly apart is what makes the enlargement transformation both powerful and predictable.