The perpendicular bisector of AB passes through its midpoint at 90. Every point on this line is equally distant from A and B; conversely, a point equally distant from the two endpoints lies on it. Open the compasses to more than half of AB. Keeping the radius unchanged, draw arcs from A and B crossing at P and Q. Draw the straight line PQ. Do not erase the arcs. Equal radii give PA=PB and QA=QB. The two arc intersections locate the equal-distance line. This explains why an arbitrary line through the midpoint is insufficient: it must also be perpendicular. Example For AB=6 cm, a compass radius of 3.6 cm gives two intersections. The constructed midpoint has AM=MB=3 cm. A radius of only 3 cm makes the arcs touch once and does not locate the bisector line.

The perpendicular bisector of \(AB\) passes through its midpoint at \(90^\circ\). Every point on this line is equally distant from \(A\) and \(B\); conversely, a point equally distant from the two endpoints lies on it.

Open the compasses to more than half of \(AB\). Keeping the radius unchanged, draw arcs from \(A\) and \(B\) crossing at \(P\) and \(Q\). Draw the straight line \(PQ\). Do not erase the arcs.

Equal radii give \(PA=PB\) and \(QA=QB\). The two arc intersections locate the equal-distance line. This explains why an arbitrary line through the midpoint is insufficient: it must also be perpendicular.

For \(AB=6\text{ cm}\), a compass radius of \(3.6\text{ cm}\) gives two intersections. The constructed midpoint has \(AM=MB=3\text{ cm}\). A radius of only \(3\text{ cm}\) makes the arcs touch once and does not locate the bisector line.

Perpendicular bisectors

Perpendicular bisectors