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 } \neq 0 $
$\overrightarrow{ PQ }\|\overrightarrow{ SR }, \overrightarrow{ PS }\| \overrightarrow{ QR } $ and $|\overrightarrow{ PQ }|=|\overrightarrow{ SR }|,|\overrightarrow{ PS }|=\mid \overrightarrow{ QR }$
Hence, $PQRS$ is a parallelogram but not rhombus or rectangle.