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)

parallelogram, which is neither a rhombus nor a rectangle

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