Question

$P,Q,R,S$ have position vectors $\overline{p},\overline{q},\overline{r},\overline{s}$ respectively such that $\overline{p}-\overline{q}=2(\overline{s}-\overline{r})$, then which of the following is correct

• 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

Given : $P,Q,R,S$ have position vectors $\overline{p},\overline{q},\overline{r},\overline{s}$

$\overline{p}-\overline{q}=2(\overline{s}-\overline{r})$

$⟹\overline{p}-\overline{q}=2\overline{s}-2\overline{r}$

$⟹\overline{p}+2\overline{r}=\overline{q}+2\overline{s}$

$⟹\frac{\overrightarrow{p}+2\overrightarrow{r}}{3}=\frac{\overrightarrow{q}+2\overrightarrow{s}}{3}$

$⟹\frac{\overrightarrow{P}+2\overrightarrow{R}}{3}=\frac{\overrightarrow{Q}+2\overrightarrow{S}}{3}$
Thus, $QS$ & $PR$ trisect each other.