Find straight distances using the right triangle in a diagram
Read horizontal and vertical separations from a diagram · Use Pythagoras for a diagonal distance · Distinguish straight distance from a path along grid lines
Section Paper 4 · Historical Extended · 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
- Right triangles and square roots
- Coordinate reading
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
- Read horizontal and vertical separations from a diagram
- Use Pythagoras for a diagonal distance
- Distinguish straight distance from a path along grid lines
Use the supplied diagram
The figures use equal horizontal and vertical unit lengths. Read the horizontal and vertical separations, form the indicated right triangle, then use straight distance = √(horizontal separation²+vertical separation²). This is a non-negative length. A purely horizontal or vertical segment needs only one coordinate difference.
For the example A(−2,−1), B(1,3), the right-triangle legs are 3 and 4. The dotted route supplies the legs; the blue diagonal is the requested straight segment.
Do not add the legs for a straight distance
Adding the horizontal and vertical lengths measures an L-shaped journey rather than the diagonal. Reversing the endpoint order changes the signed coordinate differences but not the distance. A point to itself has distance zero. Each practice question has its own diagram below it, as support for reading the geometry.
Sources for this lesson
Pause and explain
Why are the dotted horizontal and vertical legs useful?
Worked example
Find AB in the example diagram, A(−2,−1), B(1,3).
Show the worked solution
- Horizontal separation is 3 and vertical separation is 4.
- AB=√(3²+4²)=5.
Answer 5 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.
Use the diagram below to find EF, with E(−1,0) and F(2,4).
Give me a hint
Read the two perpendicular legs.
Compare my reasoning
- Horizontal separation=3; vertical separation=4.
- EF=√(9+16)=5.
5 units.
Look for these in your work
- Use the provided right triangle
- Take the non-negative root
In the next diagram, find PQ for P(−1,−2), Q(−1,3), and RS for R(0,2), S(3,2).
Give me a hint
Each segment changes only one coordinate.
Compare my reasoning
- PQ is vertical with length 3−(−2)=5.
- RS is horizontal with length 3−0=3.
PQ=5 units; RS=3 units.
Look for these in your work
- Identify the segment direction
- Use the changing coordinate
In the final diagram, T(0,0) and U(3,4) are joined. A student adds 3+4 and reports distance 7. Find TU and explain which journey has length 7.
Give me a hint
Compare the diagonal with the dotted route.
Compare my reasoning
- TU=√(3²+4²)=5.
- The dotted horizontal-then-vertical journey has length 7.
TU=5 units; the grid-line journey is 7 units.
Look for these in your work
- Distinguish path and straight distance
- Use the shown right triangle
Common mistakes
- Adding the two legs does not give diagonal length.
- A distance is non-negative even when coordinates are negative.
What can you explain now?
Why are the dotted horizontal and vertical legs useful?
Compare with the explanation
They form a right triangle whose hypotenuse is the straight distance.
Pythagoras combines perpendicular displacements to obtain the diagonal length.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Why are the dotted horizontal and vertical legs useful?
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.