Question

When constructing a perpendicular bisector, why must the compass opening be greater than 1/2 the length of the segment?

Answer 1

if the distance is less than 1/2, the 2 circles don't intersect
because if the compass length is less than half the length of the segment the two arcs of the compass will never intersect. Try it for yourself and see what happens.

Answer 2

i would say it doesnt have to be ...

Answer 3

if the compass length is less then 1/2, then you simply move the compass to the places where the arcs cross the segment, and rescribe.

Answer 4

eventually, the arcs will cross each other ...