When constructing a perpendicular bisector, why must the compass opening be greater than one half the length of the segment?

Answer.
It is Important to open the compass more than half the length so that the arcs made would meet.

Explanation.
When you want to construct a perpendicular line to a line segment, say line AB, the follow steps are taken;
1) Open your compass to a radius greater than half the line AB.
2) With A as your center make an arc on top and below line AB.
3) Using the same radius and with B as our center, make other arcs on top and below line AB to meet the first arcs on both sides of the line.
4) Join the two intersection of the two arcs to cut line AB at M.
In so doing the line AB is said to be bisected perpendicularly. If the compass was opened less than half line AB, then the arcs could not have meet.