Bisectors and geometric construction

A perpendicular bisector describes equal distances

The perpendicular bisector of segment AB passes through its midpoint at 90°. Every point on it is equally distant from A and B.

To construct it, open the compasses to more than half AB. Keeping the same radius, draw arcs centred on A and B that intersect above and below the segment. Join the two intersections. Each intersection is equally far from A and B, so their joining line is the perpendicular bisector. A radius below half AB gives no intersections; exactly half gives only the midpoint and does not determine the line this way.

Bisect an angle with matched arcs

From the vertex O, draw an arc crossing the two rays at P and Q. With equal radii centred on P and Q, draw two arcs meeting inside the angle at R. Join O to R. The equal lengths make matching triangles OPR and OQR, so OR divides the original angle into equal angles.

Points on an angle bisector are equally distant from its two side lines, using perpendicular distances. Distances to the original ray endpoints are a different comparison.

Check your understanding 1 Why must the radius stay the same?

In the perpendicular-bisector construction, a learner uses a 4 cm radius from A and a 5 cm radius from B. Explain why joining the arc intersections does not establish the required bisector.

Your answer or drawing on paper

Show answer

The intersections are 4 cm from A and 5 cm from B, so they do not have the equal-distance property. Equal radii are essential. Use one unchanged radius greater than half AB.

What this page can check

Construct a 60° angle or a triangle from your own valid side lengths on paper. State which equal lengths or angles your method guarantees, then measure as a check. The page’s written solution cannot certify the accuracy of a physical compass, ruler or protractor construction.