Replace suitable compositions by one transformation
Recognise a half-turn followed by translation as a new half-turn · Combine parallel or intersecting reflections correctly · Establish a whole-map rule rather than one-point agreement
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
- Transformation order
- Half-turn centers and reflections
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
- Recognise a half-turn followed by translation as a new half-turn
- Combine parallel or intersecting reflections correctly
- Establish a whole-map rule rather than one-point agreement
Derive the coordinate rule
A half-turn about O sends P to −P. A following translation t gives −P+t = 2(t/2)−P, a half-turn centered at t/2. For t=[4; −2], that center is (2,−1). A half-turn around A then a half-turn around B gives P+2(B−A), a translation.
Checking one source point is insufficient to prove two maps equivalent. Derive the same rule for every (x,y), or justify the geometry for the whole plane.
Two mirrors
Reflection in x=a then x=b gives x′=2b−(2a−x)=x+2(b−a), y unchanged: a translation perpendicular to the mirrors. For intersecting mirrors, their composition is a rotation about the intersection by twice their directed angle, with order deciding its direction.
Reflection in the x-axis then y=x gives (x,y)→(x,−y)→(−y,x), a 90° counterclockwise rotation about O. A rotation mixed with a non-uniform stretch is not automatically reducible to a single rotation or reflection; check invariants and the actual rule.
Sources for this lesson
Pause and explain
A half-turn about O followed by translation t is a half-turn about which point?
Worked example
Replace a half-turn about O followed by translation [6; 4] by a single transformation.
Show the worked solution
- Combined rule is P′=−P+[6; 4].
- Write it as 2(3,2)−P.
Answer Rotation 180° about (3,2).
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.
Replace a half-turn about O followed by translation [8; −6].
Give me a hint
Take half the following translation as the new center.
Compare my reasoning
- Rule is P′=−P+[8; −6].
- The half-turn center is (4,−3).
Rotation 180° about (4,−3).
Look for these in your work
- Derive the full rule
- State center and angle
Replace reflection in x=1 followed by reflection in x=4.
Give me a hint
Apply the second reflection to the first image.
Compare my reasoning
- x→2−x→8−(2−x)=x+6.
- y stays unchanged.
Translation [6; 0].
Look for these in your work
- Respect order
- Describe the whole-plane map
Replace reflection in the x-axis followed by reflection in y=x. State how reversing these two mirrors changes the result.
Give me a hint
Write both coordinate rules.
Compare my reasoning
- First order gives (−y,x): 90° counterclockwise about O.
- Reverse order gives (y,−x): 90° clockwise about O.
90° counterclockwise about O; reversing gives 90° clockwise.
Look for these in your work
- Match the correct rotation rule
- Explain order-dependent direction
Common mistakes
- Translation after a half-turn changes its center.
- One matching point does not prove whole-map equivalence.
What can you explain now?
A half-turn about O followed by translation t is a half-turn about which point?
Compare with the explanation
t/2.
The output −P+t equals 2(t/2)−P.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
A half-turn about O followed by translation t is a half-turn about which point?
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.