Question

$P,Q,R,S$ have position vectors $p,q,r,s$, such that $p-q=2(s-r)$, then

• A. $PQ$ and $RS$ bisect each other
• B. $PQ$ and $PR$ bisect each other
• C. $PQ$ and $RS$ trisect each other
• D. $QS$ and $PR$ trisect each other

Answer 1

The correct option is D $QS$ and $PR$ trisect each other

$\overrightarrow{p}-\overrightarrow{q}=2(\overrightarrow{s}-\overrightarrow{r})\Rightarrow \overrightarrow{p}+2\overrightarrow{r}=2\overrightarrow{s}+\overrightarrow{q}=\overrightarrow{q}+2\overrightarrow{s}$
$\Rightarrow \frac{\overrightarrow{p}+\overrightarrow{2}r}{1+2}=\frac{\overrightarrow{q}+2\overrightarrow{s}}{1+2}$
$\therefore$ the point divides $PR$ in the ratio $2:1$, divides $QS$ in the ratio $2:1$.
$\therefore$ $PR$ and $QS$ trisect each-other