With only a compass and a straightedge is it possible to construct a square.?

Answer 1

yes.

First draw a line segment and construct a perpendicular bisector (do you already know how, by the way?).

Once you have this you can use the compass to mark off equal lengths perpendicular to each other at the point of intersection (two sides of your square).

Then constructing arcs from the far ends of the two sides will identify the fourth corner of the square (the arcs will intersect there).

Answer 2

Yes but you might need a pencil as well. Extend one end of line marking point from which you start. Then put compass on that point and mark in either direction on line and extende bits . This gives you two points equidistant from that point on the extended line. You can then using compass on each of these points set to any distance greater than distance of these constructed points to point at end of line construct a point which is on perpendicular to this line. Draw perpendicular. Set compass to length of original line . Mark New point on perp etc. Bit confusing in words I know.