Find perpendicular lines with the correct zero-slope exceptions
Use the negative reciprocal for non-zero finite slopes · Find a perpendicular line through a supplied point · Handle horizontal and vertical lines without invalid reciprocal arithmetic
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
- Line equations
- Quarter-turn rotations and slope
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 the negative reciprocal for non-zero finite slopes
- Find a perpendicular line through a supplied point
- Handle horizontal and vertical lines without invalid reciprocal arithmetic
A quarter-turn changes direction
A finite slope m has direction (1,m). A 90° counterclockwise turn changes it to (−m,1), whose slope is −1/m when m≠0. Thus two non-zero finite slopes of perpendicular lines multiply to −1. Taking only the reciprocal, without changing its sign, does not generally give a right angle.
Keep the given point
For a line perpendicular to y=2x+1 through (2,3), use slope −1/2 and solve 3=(−1/2)×2+b, giving b=4. Horizontal and vertical lines are perpendicular to each other: a horizontal given line needs a vertical answer x=the point's x, and a vertical given line needs a horizontal answer y=the point's y. Their relationship is not a finite-slope product calculation.
Sources for this lesson
Pause and explain
What is perpendicular to a horizontal line through point (2,−1)?
Worked example
Find a perpendicular to y=2x+1 through (2,3).
Show the worked solution
- Required slope=−1/2.
- 3=−1+b gives b=4.
Answer y=−(1/2)x+4.
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 a line perpendicular to y=3x−2 through (3,1).
Give me a hint
Use slope −1/3, then substitute the point.
Compare my reasoning
- 1=(−1/3)×3+b gives b=2.
- The point and slope both satisfy y=−(1/3)x+2.
y=−(1/3)x+2.
Look for these in your work
- Use the negative reciprocal
- Verify the point
Through (2,−1), find perpendiculars to y=4 and to x=5 respectively.
Give me a hint
Treat horizontal and vertical directions separately.
Compare my reasoning
- Perpendicular to y=4 is x=2.
- Perpendicular to x=5 is y=−1.
x=2; y=−1.
Look for these in your work
- Handle zero and undefined slopes
- Retain the supplied point
A student claims slopes 2 and −1/4 are perpendicular. Test the claim and find the perpendicular slope to 2.
Give me a hint
Check the product of the two finite slopes.
Compare my reasoning
- 2×(−1/4)=−1/2, not −1.
- The required negative reciprocal is −1/2.
The claim is false; the required slope is −1/2.
Look for these in your work
- Test the relationship
- Correct both sign and reciprocal size
Common mistakes
- Change the sign as well as taking the reciprocal.
- Do not take −1/0 for a horizontal line.
What can you explain now?
What is perpendicular to a horizontal line through point (2,−1)?
Compare with the explanation
The vertical line x=2.
A horizontal direction turns to a vertical one; that line keeps the point's x coordinate.
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Make it stick
What is perpendicular to a horizontal line through point (2,−1)?
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.