Question

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

Answer 1

The circles are centered on the end points of the line segment.
They have equal radii.
What happens if the radius is less than 1/2 between the end-points?
You get a different picture.

Answer 2

So the circles do not intersect, so you can not draw a line through the points where the circles intersect.