Question

Let $P,Q,R$ and $S$ be the points on the plane with position vectors $-2\hat{i}-\hat{j},4\hat{i},3\hat{i}+3\hat{j}$ and $-3\hat{i}+2\hat{j}$ respectively. The quadrilateral PQRS must be a

• A. parallelogram, which is neither a rhombus nor a rectangle
• B. square
• C. rectangle, but not a square
• D. rhombus, but not a square

Answer 1

Answer: (A)

Evaluating midpoint of $PR$ and $QS$ which gives $M\equiv \left[\frac{\hat{i}}{2}+\hat{j}\right]$, same for both.
$\overline{PQ}=\overline{SR}=6\hat{i}+\hat{j}$
$\overline{PS}=\overrightarrow{QR}=-\hat{i}+3\hat{j}$
$\Rightarrow \overline{PQ}\cdot \overline{PS}\ne 0$
$\overrightarrow{PQ}\Vert \overrightarrow{SR},\overrightarrow{PS}\Vert \overrightarrow{QR}$ and $|\overrightarrow{PQ}|=|\overrightarrow{SR}|,|\overrightarrow{PS}|=\mid \overrightarrow{QR}$
Hence, $PQRS$ is a parallelogram but not rhombus or rectangle.