Undo a translation, reflection or rotation
Negate a translation vector · Reuse the same reflection line · Reverse a rotation's turn at the same center
Section Paper 2 · Historical Extended · non-calculator
Exam scope and assessment notes
Extended content; not a Core requirement. 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
- Basic plane transformations
- Opposite movements
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
- Negate a translation vector
- Reuse the same reflection line
- Reverse a rotation's turn at the same center
Undo with the same geometry
A translation by v is undone by −v. A reflection is its own inverse when applied again in the same mirror. A rotation is undone by the opposite signed angle about the same center; a half-turn is its own inverse.
Check the round trip
Start with P, apply the forward map, then the proposed inverse. A valid inverse returns P for every input, not just one convenient point. Using the opposite rotation direction around a different center is not generally the inverse.
If several transformations are composed, reverse their order as well as undoing each one. That combined case is developed in the composition lessons.
Sources for this lesson
Pause and explain
What is the inverse of reflection in y=2?
Worked example
Undo translation [3; −2] from the image point (5,1).
Show the worked solution
- Inverse vector is [−3; 2].
- Add it to (5,1) to recover (2,3).
Answer Original point (2,3).
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.
Translation [−4; 5] gives image (1,7). Find the original point.
Give me a hint
Use inverse [4; −5].
Compare my reasoning
- 1+4=5.
- 7−5=2.
(5,2).
Look for these in your work
- Negate both vector components
- Check the forward image
Reflection in y=2 gives image (6,−1). Recover the original.
Give me a hint
Reflect again in the same line.
Compare my reasoning
- x stays 6.
- y=4−(−1)=5.
(6,5).
Look for these in your work
- Keep the same mirror
- Use the round-trip check
A 90° counterclockwise rotation about (1,1) gives image (2,4). Recover the original.
Give me a hint
Turn the image-relative vector clockwise.
Compare my reasoning
- Image offset is (1,3), whose clockwise image is (3,−1).
- Add (1,1) to obtain (4,0); forward rotation returns (2,4).
(4,0).
Look for these in your work
- Reverse the turn
- Preserve the center
Common mistakes
- Keep the same mirror or rotation center.
- An inverse must work as a round trip for all inputs.
What can you explain now?
What is the inverse of reflection in y=2?
Compare with the explanation
Reflection again in y=2.
It returns every point across the same mirror to its original position.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
What is the inverse of reflection in y=2?
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.