Use coordinate differences for exact distances without a supplied diagram
Use √((x₂−x₁)²+(y₂−y₁)²) · Retain exact irrational distances · Compare a point-to-center distance with a circle radius
Section E7.2 · Extended · Distances between points
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
- Diagram-supported distances
- Squares and roots
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 √((x₂−x₁)²+(y₂−y₁)²)
- Retain exact irrational distances
- Compare a point-to-center distance with a circle radius
Generalise the perpendicular legs
In Cartesian coordinates with the same unit on both axes, distance between A and B is √((x₂−x₁)²+(y₂−y₁)²). The differences can be signed because they are squared. This formula expresses the same right-triangle method even when the problem supplies coordinates without a diagram.
For A(−4,−1), B(2,7), the changes are 6 and 8, giving distance 10. Reverse the endpoints and both changes reverse sign, but the distance remains the same.
Use the correct comparison
Keep an exact root such as √13 if no decimal is requested. Distance zero means the two points coincide. For a circle, compare the distance to its center with the radius: equal is on the circumference, smaller is inside and larger is outside. Both coordinate axes must represent the same length unit for this direct formula.
Sources for this lesson
Pause and explain
Does reversing both endpoints change the distance?
Worked example
Find distance from (−4,−1) to (2,7).
Show the worked solution
- Differences are 6 horizontally and 8 vertically.
- Distance=√(36+64)=10.
Answer 10 units.
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.
Find the distance from (−3,2) to (1,5).
Give me a hint
Use changes 4 and 3.
Compare my reasoning
- Distance²=4²+3²=25.
- Distance=5.
5 units.
Look for these in your work
- Use endpoint differences
- Select the non-negative root
Find the exact distance between (−2,1) and (1,3). What is the distance in the reverse direction?
Give me a hint
The changes are 3 and 2.
Compare my reasoning
- Distance=√(9+4)=√13.
- Reversing the changes does not alter their squares.
√13 units in either direction.
Look for these in your work
- Keep an exact root
- Check reversal
A circle has center (1,−1) and radius 5 units. Classify points P(4,3) and Q(5,3) as on, inside or outside it.
Give me a hint
Compare each center distance with 5.
Compare my reasoning
- P has changes (3,4), so its distance is 5 and it is on the circumference.
- Q has changes (4,4), so distance √32 exceeds 5 and it is outside.
P is on the circumference; Q is outside.
Look for these in your work
- Use the center, not the origin
- Compare squared distances or exact lengths
Common mistakes
- Square the differences, not the original coordinates.
- Do not add the two absolute changes for straight distance.
What can you explain now?
Does reversing both endpoints change the distance?
Compare with the explanation
No; both squared differences stay the same.
Each signed difference is negated, which leaves its square unchanged.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Does reversing both endpoints change the distance?
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.