Enlargement and scale factors | Year 9

01 / LEARN

Understand the idea.

Two pieces of information

An enlargement is specified by a centre and a scale factor. For a positive factor $k$ , each image point lies on the ray from the centre through its original point, with its distance from the centre multiplied by $k$ .

With centre at the origin, this becomes $(x, y) \mapsto (k x, k y)$ . A positive factor greater than one increases lengths; a factor between zero and one reduces them. The centre stays fixed, and $k = 1$ leaves every point unchanged.

Equal coordinate scales show triangle A at one comma one, B at three comma one and C at one comma two, and its enlargement A prime at two comma two, B prime at six comma two and C prime at two comma four. The centre is O at the origin and the stated scale factor is two.

A centre away from the origin

For another centre, scale the displacement rather than the coordinates. Subtract the centre coordinates from the point, multiply the resulting pair by $k$ , then add the centre back. This keeps the centre fixed throughout the transformation.

Positive enlargement preserves angles and shape, while multiplying corresponding lengths and perimeter by $k$ . Adding the same coordinate pair to every point instead produces a translation: it moves a figure without changing its size.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Origin-centred enlargement

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

Worked solution

1
Use the origin-centred coordinate rule.
$P (x, y) \mapsto P^{'} (2 x, 2 y)$
The scale factor is two and the centre is the origin.
2
Double both signed coordinates.
$\begin{aligned} x^{'} & = 2 (- 2) = - 4 \\ y^{'} & = 2 (3) = 6 \end{aligned}$
Scaling a displacement doubles each component while preserving its sign.
3
Write the image point.
$P^{'} = (- 4, 6)$
The image lies twice as far along the ray from the origin through P.

Answer

$P' = (- 4, 6)$

EXAMPLE 02

Use a different centre

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

Worked solution

1
Find the displacement from C to P.
$(4 - 1, 3 - 1) = (3, 2)$
Enlargement measures positions relative to its centre.
2
Double this displacement.
$2 (3, 2) = (6, 4)$
The scale factor applies to the displacement, not directly to the original coordinates.
3
Return to coordinates measured from the origin.
$P^{'} = (1 + 6, 1 + 4) = (7, 5)$
Adding the centre places the scaled displacement back at C.

Answer

$P' = (7, 5)$

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$A' = (3, 4)$

The centre is the origin, so halve each coordinate: $A^{'} = (\frac{6}{2}, \frac{8}{2}) = (3, 4)$ . The image is halfway from the origin to A.

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Enlarge $(2, 5)$ by factor 3 about the origin.
2 marks
Reduce $(- 8, 4)$ by factor $\frac{1}{4}$ 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 $A B C$ 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 $\frac{1}{4}$ about the origin.
2 marks
$(\frac{- 8}{4}, \frac{4}{4}) = (- 2, 1)$ .
$(- 2, 1)$
For an origin-centred enlargement, $(3, 2)$ maps to $(12, 8)$ . Find the scale factor.
2 marks
$\frac{12}{3} = 4$ and $\frac{8}{2} = 4$ .
$4$
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 $A B C$ 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.

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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 ↗

In this lesson

Understand the idea.

Two pieces of information

A centre away from the origin

See the method in action.

Origin-centred enlargement

Use the origin-centred coordinate rule.

Double both signed coordinates.

Write the image point.

Use a different centre

Find the displacement from C to P.

Double this displacement.

Return to coordinates measured from the origin.

Try it yourself.

Check your understanding

Keep practising