Question

The quadrilateral formed by joining the mid-points of the sides of a quadrilateral ABCD, taken in order, is a rhombus, if ______.

Options

• ABCD is a rhombus

• ABCD is a parallelogram

• AC = BD

• AC ⊥ BD

Answer 1

The quadrilateral formed by joining the mid-points of the sides of a quadrilateral ABCD, taken in order, is a rhombus, if AC = BD.

Explanation:

By Varignon’s theorem, the quadrilateral formed by joining the midpoints is a parallelogram whose consecutive sides equal half the lengths of the diagonals of ABCD one side = `1/2` AC, the next = `1/2` BD.

Hence, those sides are equal, so the parallelogram is a rhombus precisely when AC = BD.