ACARA v9 CONTENT DESCRIPTION “apply the enlargement transformation to shapes and objects using dynamic geometry software as appropriate; identify and explain aspects that remain the same and those that change”
Builds on: Why Trig Ratios Stay Constant: Similarity (AC9M9SP01). This unit builds on similarity and scale factor, deepening them into a transformation. The enlargement transformation is the geometric foundation for scale drawings, maps and the proportional reasoning used throughout mathematics and design.
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.
Enlarging from a centre
Rays drawn from the centre of enlargement through each vertex carry on to the image, which sits twice as far out under a scale factor of two.
The centre O is fixed and every vertex moves out along the ray from O. Under scale factor 2 each image vertex is exactly twice as far from O as the original, so the image is a larger, similar copy of the triangle.
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 mapping rule
The image of a point lies on the ray from the centre, at the scale factor times the original distance; with factor two the point at one comma one maps to two comma two.
P at (1, 1) maps to P' at (2, 2): both lie on one ray from O, and OP' = 2 × OP. The rule is always the same: the image sits at the scale factor times the original distance, measured along the ray from the 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.
Enlarge, reduce, or keep
A factor above one enlarges, a factor between zero and one reduces, and a factor of exactly one leaves the shape unchanged.
One shape at three scale factors: k = 2 enlarges it, k = 1 leaves it exactly as it was, and k = 1/2 shrinks it to a reduction. The scale factor alone decides whether the image grows, stays, or shrinks.
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.
Lengths multiply by k
Every length is multiplied by the scale factor, so a five centimetre side becomes fifteen under a factor of three, and the perimeter triples too.
Under k = 3 the five centimetre side becomes 5 × 3 = 15 centimetres, and the perimeter triples as well. Every length simply multiplies 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.
Area multiplies by k squared
When lengths double, area quadruples: a unit square enlarged by two becomes a two-by-two block of four, so area scales by the factor squared, not the factor.
Enlarging the unit square by k = 2 makes a 2 by 2 block of four unit squares, so the area goes from 1 to 4 = k². Doubling the lengths multiplies the area by four, not by two: area uses the scale 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.
Same and changed
An enlargement leaves angles, shape, parallelism and internal ratios untouched, while side lengths, perimeter, area and position all change.
The invariants on the left stay fixed: angles, shape, parallelism and internal ratios. The quantities on the right change: length × k, perimeter × k, area × k², and position. Keeping the two lists apart is the key skill.
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.
Quick self-check
1. A point at (3, 1) is enlarged from the origin with scale factor 2. Where does it map to?
2. A shape is enlarged by a scale factor of 3. A side that was 5 cm becomes:
3. A shape of area 6 is enlarged by a scale factor of 2. What is the new area?
4. Under an enlargement, which of these stays exactly the same?
5. An enlargement uses a scale factor of one half. The image is: