Negative enlargement factors
Place image points on the opposite ray and scale distance by the factor's magnitude.
Section Paper 4 · Extended · Calculator
Extended only. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Use centre-based positive enlargement.
- Multiply signed coordinates.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does the negative sign of k control?
What you will learn
- Construct enlargement with a negative scale factor.
- Separate direction reversal from length and area scaling.
A negative factor changes the ray
For centre O, the image displacement is k times the original displacement even when k is negative. With centre (1,1), point (3,2) has displacement (2,1). Factor −2 gives displacement (−4,−2) and image (−3,−1), on the opposite side of O.
Distances multiply by |k|, because physical length is non-negative. Angles stay equal and areas multiply by k². Factor −2 therefore doubles lengths and quadruples area; it does not create negative lengths or negative area.
Trace a line through the centre
Draw each vertex's line through the centre, then put the image on the opposite ray at the scaled distance. A factor between −1 and zero produces a smaller image across the centre. A factor of −1 is the same point mapping as a 180° rotation about the centre.
For a planar shape, negative enlargement is equivalent to positive enlargement by |k| followed by a half-turn, so orientation is preserved in the rotation sense; it is not a mirror reflection. Give the signed factor and centre in the transformation description.
Pause and explain
Factor −2 is applied to a shape. What is its area multiplier?
Worked example
Enlarge (3,2) about (1,1) by factor −2.
Show the worked solution
- Subtract the centre to get displacement (2,1).
- Scale to (−4,−2).
- Add the centre back to obtain (−3,−1).
Answer (−3,−1).
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Enlarge (4,−2) by factor −1/2 about the origin.
Give me a hint
Multiply both displacement components by negative one half.
Compare my reasoning
- x′ = −2.
- y′ = 1.
- The image is (−2,1), half as far away on the opposite ray.
(−2,1).
Look for these in your work
- Reversed the ray.
- Used the fractional distance multiplier.
Enlarge A(2,3) about C(1,1) by factor −3.
Give me a hint
Scale A − C, then add C back.
Compare my reasoning
- Displacement is (1,2).
- The image displacement is (−3,−6).
- A′ = (−2,−5).
(−2,−5).
Look for these in your work
- Used centre-relative coordinates.
- Applied the signed factor to both components.
A 5 cm side and 20 cm² area are enlarged by factor −3/2. Find image length and area.
Give me a hint
Lengths use magnitude; areas use the square.
Compare my reasoning
- Length multiplier is 1.5, giving 7.5 cm.
- Area multiplier is 2.25, giving 45 cm².
- The negative sign determines opposite placement, not negative dimensions.
7.5 cm; 45 cm².
Look for these in your work
- Separated placement from measurement.
- Squared the factor for area.
Common mistakes
- Putting a negative-factor image on the same ray.
- Reporting negative area.
- Calling it a reflection without a mirror.
What can you explain now?
What familiar mapping equals enlargement factor −1?
Compare with the explanation
A 180° rotation about the same centre.
Both negate every displacement from that centre without changing distance.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, compare factors 1/2 and −1/2 about the same centre.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.