Enlargement and scale factors | Year 9

01 / LEARN

The idea, step by step

Two pieces of information

An enlargement needs a centre and a scale factor. Every point moves along the line through the centre and that point. Its new distance from the centre is multiplied by the scale factor.

For a centre at the origin and positive scale factor $k$ , multiply both coordinates: $(x, y) \mapsto (k x, k y)$ . A factor between zero and one makes a reduction, although the transformation is still called an enlargement.

A centre away from the origin

Do not multiply coordinates directly unless the centre is the origin. From centre $C$ , find the displacement to the original point, multiply that displacement by $k$ , and add it back to $C$ .

Angles and shape are preserved. Corresponding lengths and the perimeter are multiplied by $k$ . A translation adds the same displacement to every point; an enlargement multiplies distances from one centre.

A common trap

An enlargement by factor 2 always adds 2 to each coordinate.

Instead: That would be a translation. For an origin-centred enlargement multiply both coordinates by 2; for another centre, scale the displacement from that centre.

02 / WORKED EXAMPLES

See the thinking

Example 1

Origin-centred enlargement

Enlarge $P (- 2, 3)$ by scale factor 2 about the origin.

Multiply the x-coordinate by 2: $- 2 \times 2 = - 4$ .
Multiply the y-coordinate by 2: $3 \times 2 = 6$ .

P' = (- 4, 6)

Example 2

Use a different centre

Enlarge $P (4, 3)$ by scale factor 2 about $C (1, 1)$ .

The displacement from C to P is $(4 - 1, 3 - 1) = (3, 2)$ .
Double it to $(6, 4)$ .
Add the centre: $(1 + 6, 1 + 4) = (7, 5)$ .

P' = (7, 5)

Check your understanding

Reduce $A (6, 8)$ by scale factor $1 / 2$ about the origin.

Show answer

$A' = (3, 4)$

Multiply each coordinate by $1 / 2$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Enlarge $(2, 5)$ by factor 3 about the origin.
2 marks
Reduce $(- 8, 4)$ by factor $1 / 4$ about the origin.
2 marks
For an origin-centred enlargement, $(3, 2)$ maps to $(12, 8)$ . Find the scale factor.
2 marks
A shape has perimeter 18 cm. It is enlarged by factor 1.5. Find its new perimeter.
2 marks
Find the image of $P (5, 2)$ under factor 3 about $C (2, 1)$ .
3 marks
In the supplied diagram, name the centre and scale factor taking ABC to its image.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Enlarge $(2, 5)$ by factor 3 about the origin.
2 marks
$(2 \times 3, 5 \times 3) = (6, 15)$ .
$(6, 15)$
Reduce $(- 8, 4)$ by factor $1 / 4$ about the origin.
2 marks
$(- 8 / 4, 4 / 4) = (- 2, 1)$ .
$(- 2, 1)$
For an origin-centred enlargement, $(3, 2)$ maps to $(12, 8)$ . Find the scale factor.
2 marks
$12 / 3 = 4$ and $8 / 2 = 4$ .
4
A shape has perimeter 18 cm. It is enlarged by factor 1.5. Find its new perimeter.
2 marks
Every side is multiplied by 1.5, so perimeter $= 18 \times 1.5 = 27$ cm.
27 cm
Find the image of $P (5, 2)$ under factor 3 about $C (2, 1)$ .
3 marks
Displacement is $(3, 1)$ .
Scaled displacement is $(9, 3)$ .
Add C to get $(11, 4)$ .
$(11, 4)$
In the supplied diagram, name the centre and scale factor taking ABC to its image.
3 marks
Each image point lies on the same ray from O as its original.
Both coordinates and each distance from O are doubled.
Centre O(0,0), scale factor 2.

Checking saved progress…

Continue learning

Practical proportional models and limitations

Next lesson

Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Year 9 proportional modelling, rates, scale, similarity and enlargement. Assumptions, interpretation and model limitations are taught alongside calculation.

Official curriculum source ↗

Back to Ratio, rates and similarity