Enlarge or reduce from a center using a stated scale factor
Scale each displacement from the center · Describe center and length factor · Handle an explicitly defined signed coordinate map
Section Paper 3 · Historical Core · calculator
Exam scope and assessment notes
Core and Extended foundations. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Ratios and similar figures
- Coordinate differences
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Scale each displacement from the center
- Describe center and length factor
- Handle an explicitly defined signed coordinate map
Scale from the center, not from the origin by default
A positive dilation about C sends P to P′=C+k(P−C), with k>0. A factor greater than 1 enlarges; 0<k<1 reduces. The center is fixed, angles are unchanged and corresponding lengths multiply by k.
For a non-identity dilation, corresponding-point lines meet at the center; their center-distance ratio gives the factor. A complete line through the center maps to itself as a set, though most of its points move. A line not through the center maps to a parallel line.
Follow the factor that is stated
For an explicitly defined signed map P′=C+k(P−C), a negative k puts the image on the opposite ray, with length factor |k|. At k=−1 this is simply a 180° rotation about C. Always use the rule, factor and center stated in the question.
At k=1 the map is the identity and its center cannot be uniquely inferred from the images. At k=0 all points collapse to C; this is not a similarity of a non-degenerate figure and has no unique inverse across the plane.
Sources for this lesson
Pause and explain
What is fixed under a dilation about C?
Worked example
Enlarge P(2,0) by positive factor 2 about C(1,−1).
Show the worked solution
- Relative displacement is (1,1).
- Double it to (2,2), then add C.
Answer P′(3,1).
Try a topic test
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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 P(1,3) by factor 2 about O.
Give me a hint
Scale both origin-relative coordinates.
Compare my reasoning
- Relative vector is (1,3).
- Doubling gives (2,6).
P′(2,6).
Look for these in your work
- Scale both coordinates
- Use the given center
A positive dilation about C(1,−1) sends P(5,3) to P′(3,1). Find its factor and the image of Q(3,1).
Give me a hint
Compare relative displacements.
Compare my reasoning
- P−C=(4,4), P′−C=(2,2), so k=1/2.
- Q−C=(2,2); half gives (1,1), so Q′=(2,0).
Factor 1/2; Q′(2,0).
Look for these in your work
- Describe the factor in a consistent direction
- Verify on another point
A question defines P′=C−(P−C), with C(1,−1) and P(4,2). Find the image and identify the basic transformation.
Give me a hint
This stated signed rule has k=−1.
Compare my reasoning
- P′=2C−P=(2−4,−2−2)=(−2,−4).
- Reversing every center-relative displacement without changing length is a half-turn.
P′(−2,−4); rotation 180° about (1,−1).
Look for these in your work
- Follow the defined coordinate rule
- Identify its geometric meaning
Common mistakes
- Scale the displacement from the center.
- A zero collapse does not preserve a figure's shape.
What can you explain now?
What is fixed under a dilation about C?
Compare with the explanation
The center C.
Its relative displacement is zero, so scaling leaves it unchanged.
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Make it stick
What is fixed under a dilation about C?
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.