Is it possible to construct a perpendicular bisector to any given line segment using only a straightedge and a compas?

Answer 1

Yes

Set the compass to wider than half the length of the line segment.

Put the point of the compass on one end of the line segment and draw two arcs, one either side of the line (roughly near the middle).

Put the point of the compass on the other end of the line segment and draw two further arcs to intersect the first two arcs.

With a straight edge, join the two points where the arcs cross. This line is the perpendicular bisector of the original line segment.