Use the fact that the diagonals in squares are congruent, perpendicular, and they bisect each other.
See solution.
Practice makes perfect
We are asked to construct a square using the diagonals. Let's recall that the diagonals in squares are congruent, perpendicular, and they bisect each other. The first step in this construction will be to draw the first diagonal and name it AC.
Now, we will construct a line perpendicular to this segment that also passes through its midpoint. To do this, we will place the compass at each of the endpoints of this segment and draw arcs above and below this segment using a setting greater than a half of the segment's length.
Next, we will connect the points of intersection of the arcs with a line. This line is perpendicular to the drawn diagonal and bisects it.
To find the next two vertices, we will place the compass at the point of intersection of the line and segment, and we will adjust the compass setting so that the pencil is at point A.
Using that setting, let's turn around the compass and draw arcs that intersect the line both above and below the segment AC. Label these points of intersection B and D.
Our last step will be to connect all four vertices of our square.
