Negative enlargement factors
Place image points on the opposite ray and scale distance by the factor's magnitude.
Extended only.
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).
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.
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Number contains 19 sequenced lessons; Algebra and graphs 21; Coordinate geometry 10; Geometry 15; Mensuration 13; Trigonometry 13; Transformations and vectors 11; Probability 7; Statistics 13. All 72 syllabus section entries now link to teaching and staged practice. Seven mixed checkpoints sample skills and suggest review lessons. Nine earlier overviews remain available. Full cumulative assessment and human teacher review remain pending. Written lesson practice, charts and constructions are self-checked, not automatically graded. Checkpoints do not save an exam grade or certify mastery.
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