Construct a perpendicular through an on-line or off-line point
Create an equidistant endpoint pair on a line · Construct a perpendicular through a specified point · Avoid a tangent arc that supplies only one point
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
- Perpendicular-bisector construction
- Equal-distance reasoning
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
- Create an equidistant endpoint pair on a line
- Construct a perpendicular through a specified point
- Avoid a tangent arc that supplies only one point
Prepare your paper
When a practice task gives numerical input geometry rather than a printed figure, first prepare that source segment, angle, triangle or circle with measuring tools as needed. Then put the measuring scale or protractor aside for the requested formal copying or bisector steps: use a compass and unmarked straightedge to locate the new points. The preparation supplies the givens; it is not a replacement for the formal construction.
The website diagrams explain relationships; their screen size is not a centimetre calibration or a ready-to-measure template. Keep the input figure and useful construction arcs on paper. Your drawing is self-checked and needs human review for reliable marking.
Point on the line
For P on a line, use a P-centered arc to mark A and B on opposite sides at equal distances. Construct the perpendicular bisector of AB with equal arcs from A and B. Since P is already its midpoint, that bisector is the required perpendicular through P.
Point off the line
For P off the line, choose a P-centered circle large enough to cross it twice, at A and B. Since PA = PB, P lies on the perpendicular bisector of AB. Construct that bisector from equal endpoint arcs; it passes through P and meets the original line at right angles.
A circle merely tangent to the line gives only one intersection and cannot supply the required endpoint pair. Draw arcs large enough to give distinct points, and preserve the original line and the relevant intersections.
Sources for this lesson
Pause and explain
Why does the off-line point P lie on the bisector of AB?
Worked example
Explain a compass-and-straightedge perpendicular from P outside a line.
Show the worked solution
- Draw a P-centered arc cutting the line at two distinct points A and B.
- Construct the perpendicular bisector of AB.
- PA = PB puts P on that bisector, so the resulting line is perpendicular and passes through P.
Answer A perpendicular through P, justified by the equal-distance locus.
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.
Construct a perpendicular through P on a given line. Describe how A and B are chosen.
Give me a hint
Make them equidistant from P on opposite sides.
Compare my reasoning
- A P-centered circle marks A and B on the line.
- The perpendicular bisector of AB passes through its midpoint P.
Use symmetric A/B intersections and their perpendicular bisector.
Look for these in your work
- Locate the pair from P
- Retain the bisector arcs
Construct a perpendicular from P off a line. State the condition on the first P-centered arc.
Give me a hint
The line must cut the circle twice.
Compare my reasoning
- Choose an arc radius greater than the perpendicular distance from P to the line.
- With two intersections A/B, construct their perpendicular bisector through P.
The first circle must meet the line at two distinct points.
Look for these in your work
- Avoid a tangent-only intersection
- Use the equal-distance justification
An off-line construction finds two points A/B with unequal distances from P, then joins their midpoint to P. Is a right angle guaranteed?
Give me a hint
The equal-distance property is essential.
Compare my reasoning
- No: P may not lie on the perpendicular bisector when PA ≠ PB.
- Repair the endpoint selection using one P-centered compass circle, then construct the bisector properly.
No guarantee. Use equal PA/PB and the formal bisector construction.
Look for these in your work
- Identify the failed condition
- Explain a repair
Common mistakes
- One tangent intersection does not give an endpoint pair.
- A perpendicular in the wrong position does not solve the specified-point task.
What can you explain now?
Why does the off-line point P lie on the bisector of AB?
Compare with the explanation
Because PA = PB from the P-centered circle.
The perpendicular bisector contains points equidistant from the endpoints.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Why does the off-line point P lie on the bisector of AB?
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.