IGCSE Maths / Geometric constructions
Constructing the perpendicular bisector of a line segment revision notes
Edexcel International GCSE Maths (4MA1), chapter 13: Geometric constructions, specification point 4.5d. Free lesson.
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A perpendicular bisector cuts a line segment into two equal halves at right angles. Constructing it with a straight edge and compasses also gives a line whose points are all the same distance from the segment’s ends.
What a perpendicular bisector does
The words describe the line: perpendicular means at right angles, and bisect means split into two equal halves. So the line passes through the midpoint of a segment and crosses it at 90°. Every point on the perpendicular bisector is the same distance from the two ends of the segment. This makes it useful when a path must stay equally far from two points.
How to construct the bisector
Use a straight edge and compasses. Choose a width greater than half the segment so that the arcs from the two ends cross above and below it. Joining those crossings gives the required line.
Why the construction gives a right angle
At each crossing, the arcs show that P and Q are each the same distance from A as from B. So both points lie on the perpendicular bisector of AB, which passes through the midpoint M. The line PQ therefore passes through M. In the diagram’s explanation, the compass width is AB. Therefore AP, BP, AQ and BQ all equal AB. Triangle APB is equilateral, so each angle is 60°. PQ bisects the angle at P, giving angle APM = 30°. Also, angle PAM = 60°. In triangle APM, the angles add to 180°. So angle AMP = 180° − 30° − 60° = 90°. This shows PQ meets AB at a right angle.
Keep every construction arc visible
The arcs are evidence that you used compasses to construct the line rather than measuring its position. A correct-looking line with the arcs erased can score no marks. Leave all four arcs on the page. A ruler and protractor are not a substitute. For this construction, use a straight edge and compasses.
Use the equal-distance property
A and B are two trees, and a path must stay the same distance from both. The path lies on the perpendicular bisector of AB, because every point on that line is equally far from A and B. Remember: halfway along, at right angles; one compass width from both ends; join the two crossings; leave the arcs showing.