Rotate around a stated center by shifting the coordinates
Subtract the center before applying a rotation rule · Add the center back afterwards · Recognise the fixed center and preserved radius
Section E5.6 · Extended · Plane transformations
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
- Origin rotation rules
- Signed 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
- Subtract the center before applying a rotation rule
- Add the center back afterwards
- Recognise the fixed center and preserved radius
Center, turn, restore
For center C(h,k), first form the relative vector (x−h,y−k), rotate that vector, then add (h,k). For a 90° counterclockwise turn, the image is (h−(y−k),k+(x−h)). Applying an origin rule directly to the point ignores the stated center.
Describe and check
The center stays fixed, and every point keeps its distance from it. For a half-turn, P′=2C−P; equivalently C is the midpoint of every original/image point pair. Check more than one pair when identifying a whole-figure rotation.
The diagram's center is K(1,1): A(2,1), B(4,1), C(2,2) turn 90° counterclockwise to (1,2), (1,4), (0,2). Use the stated center for every vertex.
Sources for this lesson
Pause and explain
What must happen before using an origin rotation rule for center (h,k)?
Worked example
Rotate P(3,1) 90° clockwise about center (1,1).
Show the worked solution
- Relative vector is (2,0).
- Its clockwise image is (0,−2); add (1,1).
Answer P′(1,−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.
Rotate P(2,1) 90° counterclockwise around (1,1).
Give me a hint
Rotate the relative displacement (1,0).
Compare my reasoning
- It becomes (0,1).
- Add the center to get (1,2).
P′(1,2).
Look for these in your work
- Use the actual center
- Restore absolute coordinates
Half-turn P(3,4) about (1,2). Verify the midpoint property.
Give me a hint
Use P′=2C−P.
Compare my reasoning
- P′=(2−3,4−4)=(−1,0).
- Midpoint of (3,4) and (−1,0) is (1,2).
P′(−1,0); midpoint (1,2).
Look for these in your work
- Negate relative displacement
- Check the center
A claimed 90° rotation about (1,1) sends that center to (2,2). Can the claim be correct? Explain.
Give me a hint
Check the fixed-center condition before other points.
Compare my reasoning
- The center's relative displacement is zero.
- Any rotation around it keeps it at (1,1), so the stated image contradicts the claim.
No; the stated center must stay fixed at (1,1).
Look for these in your work
- Test an invariant point
- Reject inconsistent whole-figure descriptions
Common mistakes
- Do not forget either coordinate shift.
- The origin is not automatically the center.
What can you explain now?
What must happen before using an origin rotation rule for center (h,k)?
Compare with the explanation
Subtract (h,k) to form the relative coordinates.
The rotation acts on displacement from its actual center.
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Make it stick
What must happen before using an origin rotation rule for center (h,k)?
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.