Question

Prove that the line segments joining the mid-points of the adjacent sides of a quadrilateral, taken in order form a parallelogram.

Answer 1

Let ABCD be a quadrilateral and M,N,O,P be the mid points of the sides AB,BC,CD,DA respectively.

Position vectors of M,N,O,P are $\frac{\overrightarrow{a}+\overrightarrow{b}}{2},\frac{\overrightarrow{b}+\overrightarrow{c}}{2},\frac{\overrightarrow{c}+\overrightarrow{d}}{2},\frac{\overrightarrow{d}+\overrightarrow{a}}{2}$ respectively.

If we show that $\overrightarrow{MN}=\overrightarrow{PO}$ $\overrightarrow{MP}=\overrightarrow{NO}$ , then it means MNOP is a parallelogram.

$\overrightarrow{MN}=\frac{\overrightarrow{b}+\overrightarrow{c}}{2}-\frac{\overrightarrow{a}+\overrightarrow{b}}{2}=\frac{\overrightarrow{c}-\overrightarrow{a}}{2}$

$\overrightarrow{PO}=\frac{\overrightarrow{c}+\overrightarrow{d}}{2}-\frac{\overrightarrow{d}+\overrightarrow{a}}{2}=\frac{\overrightarrow{c}-\overrightarrow{a}}{2}$

$\therefore \overrightarrow{MN}=\overrightarrow{PO}\Rightarrow \overrightarrow{MN}\parallel \overrightarrow{PO}$

Similarly, we can prove that $\overrightarrow{MP}=\overrightarrow{NO}$ and $\overrightarrow{MP}\parallel \overrightarrow{NO}$

Hence, MNOP is a parallelogram.