137. The Perfect Split · Bisecting a line segment with precision

Try to break it

Drag A and B. As long as the arc radius is bigger than half AB, the two arcs cross at C and D and CD passes through the exact midpoint P at 90°. Spread A and B so far apart that AB exceeds twice the radius and the arcs no longer meet — that's the one way to break the construction.

How you build it

Construct the perpendicular bisector of AB.

• Place point A — the left endpoint of the segment you will bisect.
• Place point B to the right of A — the right endpoint. The segment AB is what you will bisect.
• Draw segment AB — click A then B. This is the segment you will bisect.
• Arc tool: click centre A, then click out to set a radius greater than half of AB. The arc swings above and below the segment.
• Arc tool: click centre B. The radius from step 4 is locked — no need to reset it. The two arcs cross at two points: call them C (above AB) and D (below AB).
• Line tool: draw a straight line through both intersection points C and D. This line is the perpendicular bisector of AB — it cuts AB at its exact midpoint P, and meets AB at a right angle (90°).

The proof, step by step

Prove that CD bisects AB at right angles.

• In ΔAPC and ΔBPC, AC = BC (both are radii of equal arcs drawn with the same compass width).
• PC = PC (Common side to both triangles).
• By SSS Congruence Rule, ΔAPC ≅ ΔBPC.
• Therefore, AP = PB (CPCT) and ∠APC = ∠BPC (CPCT).
• Since ∠APC + ∠BPC = 180° (linear pair), each angle is 90°. Thus, CD ⊥ AB.

Worked example

A line segment AB of length 14 cm is given. Its perpendicular bisector CD intersects AB at P. What is the length of AP?

The perpendicular bisector of a line segment divides it into two equal parts. Since AB = 14 cm, AP = PB = 14 / 2 = 7 cm.

• 5 cm
• 6 cm
• 7 cm — correct
• 8 cm