Reflect in diagonal mirrors without losing signs
Use reflection in y=x and y=−x · Handle a shifted diagonal line by shifting coordinates · Check mirror membership and distance symmetry
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
- Reflection as perpendicular bisection
- Signed coordinates
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
- Use reflection in y=x and y=−x
- Handle a shifted diagonal line by shifting coordinates
- Check mirror membership and distance symmetry
Two diagonal rules
Reflection in y=x swaps coordinates: (x,y) ↦ (y,x). Reflection in y=−x swaps and negates: (x,y) ↦ (−y,−x). Points on the stated mirror stay fixed; the joining segment is perpendicular to it.
A shifted diagonal
For y=x+2, work with the shifted vertical coordinate v=y−2. Reflect by swapping x and v, then undo the shift: (x,y) ↦ (y−2,x+2). The simpler y=x rule cannot be applied unchanged to this different line.
The general geometric method remains equal perpendicular distances, so use it if no memorised special-line rule applies. State the mirror equation and match original/image vertices.
Sources for this lesson
Pause and explain
What is the image of (2,5) in y=−x?
Worked example
Reflect P(2,5) in y=x and in y=−x.
Show the worked solution
- For y=x, swap to (5,2).
- For y=−x, swap and negate to (−5,−2).
Answer (5,2) and (−5,−2), respectively.
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.
Reflect (−3,4) in y=x.
Give me a hint
Swap without negating.
Compare my reasoning
- The old y becomes the new x.
- The old x becomes the new y.
(4,−3).
Look for these in your work
- Keep the original signs
- Use the correct mirror rule
Reflect (−2,5) in y=−x. Is (3,−3) fixed by that reflection?
Give me a hint
Swap and negate; then test mirror membership.
Compare my reasoning
- Image is (−5,2).
- (3,−3) satisfies y=−x, so it stays fixed.
(−5,2); (3,−3) is fixed.
Look for these in your work
- Handle both signs
- Check invariant points
Reflect (4,1) in y=x+2 and verify the midpoint lies on the mirror.
Give me a hint
Use the shifted rule.
Compare my reasoning
- Image is (1−2,4+2) = (−1,6).
- Midpoint is (1.5,3.5), satisfying 3.5=1.5+2.
(−1,6); midpoint (1.5,3.5) lies on the mirror.
Look for these in your work
- Use the actual line
- Check the midpoint condition
Common mistakes
- Swapping alone is wrong for y=−x.
- A shifted line needs a shifted rule.
What can you explain now?
What is the image of (2,5) in y=−x?
Compare with the explanation
(−5,−2).
Both swapped coordinates change sign.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
What is the image of (2,5) in y=−x?
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.