Bisect a segment using equal compass arcs
Construct a perpendicular bisector · Choose a radius that gives two intersections · Explain equal-distance and midpoint properties
Section E4.5 · Extended · Compass-and-straightedge constructions and scale drawings
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
- Copying distances
- Perpendicular lines
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
- Construct a perpendicular bisector
- Choose a radius that gives two intersections
- Explain equal-distance and midpoint properties
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.
Two intersections determine a line
For segment AB, choose a compass radius greater than half AB. Draw equal-radius arcs from A and B to meet above and below the segment at C and D. Draw the line CD with an unmarked straightedge. It meets AB at its midpoint M and is perpendicular to AB.
Both C and D are equidistant from A and B. The resulting line is the locus of points equidistant from the endpoints. Its intersection with the segment is therefore the midpoint, and symmetry gives the right angle.
Radius and evidence
If the radius is less than half AB, the arcs do not meet. At exactly half, they meet at only one point, insufficient to determine the bisector line by this construction. Keep equal openings for both endpoint arcs and retain both intersections; do not replace the method by measuring the midpoint.
Sources for this lesson
Pause and explain
Why must the arc radius exceed half the segment length?
Worked example
Bisect an 8 cm supplied segment with equal arcs of radius 5 cm.
Show the worked solution
- 5 cm exceeds half of 8 cm, so there are two intersections.
- Join those intersections; their line crosses the segment at right angles.
- The midpoint divides AB into 4 cm and 4 cm.
Answer Perpendicular bisector through the 4 cm midpoint, with two arc intersections retained.
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 bisector of a supplied 6 cm segment. State a suitable example radius and the final halves.
Give me a hint
Choose a radius greater than 3 cm.
Compare my reasoning
- A 4 cm radius gives two equal-distance intersections.
- Join them to locate the midpoint; each half is 3 cm.
For example radius 4 cm; midpoint halves 3 cm and 3 cm.
Look for these in your work
- Use equal radii
- Show two intersections and their joining line
For an 8 cm segment, compare radii 3 cm, 4 cm and 5 cm in this method.
Give me a hint
Compare each radius with half the length.
Compare my reasoning
- 3 cm gives no intersection; 4 cm gives one touching point.
- 5 cm gives two intersections and a usable bisector line.
Only 5 cm gives the two required intersections.
Look for these in your work
- Check existence of intersections
- Distinguish touching from crossing
A marker must be the same distance from two fixed points A and B. Must it lie on the segment AB? Describe all possible positions.
Give me a hint
Use the perpendicular-bisector locus.
Compare my reasoning
- Every point on the perpendicular bisector is equidistant from A and B.
- Only its midpoint lies on AB; other valid positions lie off the segment.
Any point on the perpendicular bisector, not only the midpoint.
Look for these in your work
- Distinguish the locus from one point
- Explain the equal-distance condition
Common mistakes
- A radius equal to half the segment gives only one intersection.
- Both endpoint arcs must use the same radius.
What can you explain now?
Why must the arc radius exceed half the segment length?
Compare with the explanation
To obtain two distinct intersections that determine the bisector line.
A smaller radius gives none; equality gives only one.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Why must the arc radius exceed half the segment length?
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.