JAMB 1991 · UME · Q33

One angle of a rhombus is $60^\circ$ . The shorter of the two diagonals is 8 cm long. Find the length of the longer one.

Worked solution (try it first)

The short diagonal joins the two $120^\circ$ corners and cuts the rhombus into two equilateral triangles, so each side is 8 cm.
The long diagonal bisects the $60^\circ$ angles and meets the short one at right angles.
In one right-angled quarter, the hypotenuse is 8 and the angle is $30^\circ$ , so half the long diagonal is $8\cos30^\circ = 4\sqrt3$ .
So the long diagonal is $2 \times 4\sqrt3 = 8\sqrt3$ cm, option A.