Use the fact that the diagonals in rhombi are perpendicular, and they bisect each other.
See solution.
Practice makes perfect
We are asked to construct a rhombus using the diagonals. Let's recall that the diagonals in rhombi are 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 and passing 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.
With this, we can choose any point that lies on this line to be the third vertex of this rhombus. Let's label it B.
Finally, we will find the fourth vertex D. To do this, we will place the compass at the point of intersection of the line and the segment, and we will adjust the compass setting so that the pencil is at point B.
Using that setting, let's turn around the compass and draw an arc that intersects the line on the other side of the segment AC. Label this point of intersection D.
Our last step will be to connect all four vertices of our rhombus.
