Locate a midpoint by halving both coordinate changes
Average both endpoint coordinates · Check that a proposed midpoint halves the segment · Recover an endpoint from the other endpoint and midpoint
Section Paper 1 · Historical Core · non-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
- Coordinate reading
- Halves and signed addition
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
- Average both endpoint coordinates
- Check that a proposed midpoint halves the segment
- Recover an endpoint from the other endpoint and midpoint
Halfway in each coordinate
For A(x₁,y₁), B(x₂,y₂), the midpoint has coordinates ((x₁+x₂)/2,(y₁+y₂)/2). Equivalently, start at A and take half of each change to B. Average the signed coordinates; do not replace negative values by their magnitudes.
The example diagram shows A(−2,0), B(4,2). The coordinate changes are 6 right and 2 up, so halfway is 3 right and 1 up from A. A midpoint can have fractional coordinates and need not be the origin.
Check or work backwards
A midpoint M has equal distances from A and B and lies on the connecting segment. Equal distances alone, without that location, do not identify a midpoint. If A and M are given, continue from M by the same displacement as A to M; algebraically B=2M−A. Use the practice diagrams to keep the point order clear.
Sources for this lesson
Pause and explain
Is a point midway only in its x coordinate necessarily the segment midpoint?
Worked example
Find the midpoint of A(−2,0) and B(4,2) in the example diagram.
Show the worked solution
- Average x: (−2+4)/2=1.
- Average y: (0+2)/2=1.
Answer M(1,1).
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.
The first practice diagram shows A(−2,0), B(4,2) and proposed midpoint M(1,1). Verify M using both coordinate changes.
Give me a hint
Compare A→M with M→B.
Compare my reasoning
- A→M moves (3,1).
- M→B also moves (3,1), so M is halfway along AB.
Yes: both half-segment changes are (3,1).
Look for these in your work
- Check both coordinates
- Check position on the segment
Find the midpoint of C(−3,3) and D(1,−1) in the next diagram.
Give me a hint
Average the two signed endpoint coordinates.
Compare my reasoning
- x=(−3+1)/2=−1.
- y=(3−1)/2=1.
(−1,1).
Look for these in your work
- Use both averages
- Keep the signs
The last diagram gives A(−2,−1) and midpoint M(1,2). Locate the missing endpoint B and explain the equal displacements.
Give me a hint
Continue the same movement from M.
Compare my reasoning
- A→M is (3,3), so M→B must be (3,3).
- B=(1+3,2+3)=(4,5).
B(4,5); A→M and M→B both equal (3,3).
Look for these in your work
- Respect the given midpoint
- Recover and verify the endpoint
Common mistakes
- Average both coordinates with their signs.
- A halfway x coordinate alone is not enough.
What can you explain now?
Is a point midway only in its x coordinate necessarily the segment midpoint?
Compare with the explanation
No; its y coordinate must also be halfway.
The midpoint halves the complete displacement, not just its horizontal part.
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Make it stick
Is a point midway only in its x coordinate necessarily the segment midpoint?
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.