When constructing a
perpendicular bisector, why must
the compass opening be greater
than 1/2 the length of the
segment?
The compass opening must be greater than 1/2 the length of the segment because the perpendicular bisector needs to intersect the segment at its midpoint. The midpoint is exactly half of the segment's length away from both endpoints.
By using a compass opening greater than 1/2 the length of the segment, we can create two circles centered at each endpoint of the segment. The point where these two circles intersect will be the midpoint of the segment, and drawing a line through this point will create the perpendicular bisector.
If the compass opening is smaller than 1/2 the length of the segment, the two circles will not intersect at a point that is exactly half the length away from the endpoints, resulting in an incorrect perpendicular bisector.