Verify right angles and measures using slopes and distances
Check perpendicularity with finite slopes or axis directions · Verify a right-angle claim with squared distances · Avoid applying triangle formulae to collinear data
Section Paper 2 · Historical Extended · non-calculator
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
- Distance formula
- Perpendicular slopes and triangle area
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
- Check perpendicularity with finite slopes or axis directions
- Verify a right-angle claim with squared distances
- Avoid applying triangle formulae to collinear data
Two checks should agree
For A(1,1), B(5,3), C(0,3), slopes AB=1/2 and AC=−2 have product −1, establishing a right angle at A. Independently, AB²=20, AC²=5 and BC²=25, so the converse of Pythagoras gives the same result. Use the two sides meeting at the named angle, not an unrelated slope pair.
Calculate after establishing the condition
The perpendicular legs have lengths 2√5 and √5. Area=(1/2)(2√5)(√5)=5 square units, and perimeter=2√5+√5+5=5+3√5 units. If the points are collinear there is no non-degenerate triangle, even though pairwise distances can still be calculated. A drawn angle alone is not proof.
Sources for this lesson
Pause and explain
What do finite slopes 1/2 and −2 tell you about their line directions?
Worked example
Verify the right angle and find the area of the illustrated triangle.
Show the worked solution
- The slopes from A are 1/2 and −2, with product −1; squared lengths also satisfy 20+5=25.
- Area=(1/2)√20√5=5.
Answer Right angle at A; area 5 square 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.
Pairs of distinct lines have slopes (3,−1/3) and (3,3). Classify each pair.
Give me a hint
Compare products for perpendicularity and equality for parallelism.
Compare my reasoning
- The first pair multiplies to −1, so is perpendicular.
- The second pair has equal finite slopes and distinct lines, so is parallel.
First: perpendicular; second: parallel.
Look for these in your work
- Apply the right test
- Use the distinct-line condition
For the illustrated A(1,1), B(5,3), C(0,3), verify the right angle and find area and perimeter.
Give me a hint
Use slopes and exact distances.
Compare my reasoning
- AB slope=1/2, AC slope=−2, so the right angle is at A. Lengths are √20, √5 and 5.
- Area=5 square units; perimeter=5+3√5 units.
Right angle at A; area 5 square units; perimeter 5+3√5 units.
Look for these in your work
- Establish perpendicular legs
- Keep exact measures
For A(0,0), B(2,1), C(4,2), test a claim of a right angle at B and a positive triangle area.
Give me a hint
Compare directions and the distance relation.
Compare my reasoning
- AB and BC both have slope 1/2, so are collinear, not perpendicular.
- AB=BC=√5 and AC=2√5; no enclosed triangle exists, so area=0.
The claims are false; the points are collinear and area is 0.
Look for these in your work
- Reject inconsistent angle claims
- Recognise degenerate geometry
Common mistakes
- Use the segments meeting at the angle being tested.
- A visual resemblance to 90° is not enough.
What can you explain now?
What do finite slopes 1/2 and −2 tell you about their line directions?
Compare with the explanation
They are perpendicular.
Their product is −1, meeting the non-zero finite-slope condition.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
What do finite slopes 1/2 and −2 tell you about their line directions?
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.