The idea, step by step
Scale from the centre
For positive scale factor k, an image point lies on the ray from the centre through the original point. Multiply its distance from the centre by k. Factors above 1 enlarge; factors between 0 and 1 reduce.
From the origin, multiply both coordinates by k. The original point, image and origin should line up. In the diagram, each point has scale factor 2.
Use a different centre
If the centre is not the origin, first find the displacement from the centre to the point. Scale that displacement, then add the centre back. Multiplying the position coordinates directly would use the wrong centre.
Plot the centre, point and image to check alignment and distance ratios. A digital geometry tool can check the calculation, but record the centre and factor rather than only showing a screenshot.
A common trap
Factor 2 means add 2 to each coordinate.
Instead: Adding a fixed displacement translates; enlargement multiplies distances from its centre.
See the thinking
Enlarge a triangle
Enlarge A(1,1), B(3,1), C(1,2) by factor 2 from the origin.
- Double both coordinates of every vertex.
- A′=(2,2), B′=(6,2), C′=(2,4).
- The horizontal and vertical lengths both double.
A shifted centre
Enlarge P(3,2) by factor 3 from C(1,1).
- P−C=(2,1).
- 3(P−C)=(6,3).
- C+(6,3)=(7,4).
Check your understanding
Enlarge (−4,6) by factor 1/2 from the origin.
Show answer
(−2,3)
Halve both coordinates without changing their signs.
Your turn
6 questions · 15 marks · Work at your own pace.
Read the questions online
- Enlarge P(2,−3) by factor 2 from the origin.2 marks
- Enlarge Q(−6,4) by factor 1/2 from the origin.2 marks
- Enlarge A(1,0), B(3,0), C(1,2) by factor 3 from the origin.3 marks
- Centre C is (1,−1). Enlarge P(3,2) by factor 2.3 marks
- An enlargement from the origin maps (2,3) to (6,9). Find k.2 marks
- A student maps (3,1) to (5,3) and calls it factor 2 from the origin. Give the correct image and describe the student’s movement.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Enlarge P(2,−3) by factor 2 from the origin.2 marks
2×2=4; 2×(−3)=−6.
P′(4,−6). - Enlarge Q(−6,4) by factor 1/2 from the origin.2 marks
Halve both coordinates.
Q′(−3,2). - Enlarge A(1,0), B(3,0), C(1,2) by factor 3 from the origin.3 marks
Multiply every coordinate by 3.
Both perpendicular sides triple.
A′(3,0), B′(9,0), C′(3,6). - Centre C is (1,−1). Enlarge P(3,2) by factor 2.3 marks
P−C=(2,3).
Double to (4,6), then add C to get (5,5).
P′(5,5). - An enlargement from the origin maps (2,3) to (6,9). Find k.2 marks
6÷2=3 and 9÷3=3. Both coordinates give the same factor.
3 - A student maps (3,1) to (5,3) and calls it factor 2 from the origin. Give the correct image and describe the student’s movement.3 marks
The correct image doubles both coordinates.
The student added the same displacement to the point.
Correct image (6,2); the student translated by (2,2).
Curriculum reference & lesson scope
Australian Curriculum v9 / South Australian Curriculum
Angle reasoning consolidation; enlargements using coordinates/digital geometry; invariant and changing properties; geometric constructions; sequences of steps and decisions; testing/refining construction algorithms; explaining geometric reasoning.
Official curriculum source ↗