Question

Let $\square PQRS$ be a quadrilateral. If $M$ and $N$ are midpoints of the sides $PQ$ and $RS$ respectively then $\overline{PS}+\overline{QR}=$

• A. $3\overline{MN}$
• B. $4\overline{MN}$
• C. $2\overline{MN}$
• D. $2\overline{NM}$

Answer 1

The correct option is C $2\overline{MN}$

Let $\overline{m},\overline{n},\overline{p},\overline{q},\overline{r}$ and $\overline{s}$ be position vectors of $M,N,P,Q,R$ and $S$ respectively.

$M$ and $N$ are midpoints of $PQ$ and $RS$

$\therefore \overline{m}=\frac{\overline{p}+\overline{q}}{2}$ and
$\overline{n}=\frac{\overline{r}+\overline{s}}{2}$

$\overline{PS}+\overline{QR}=\overline{s}-\overline{p}+\overline{r}-\overline{q}$
$=(\overline{r}+\overline{s})-(\overline{p}+\overline{q})$
$=2\overline{n}-2\overline{m}$

$=2(\overline{n}-\overline{m})$

$=2\overline{MN}$