Let $P, Q, R$ and $S$ be the points on the plane with position vectors $-2 \hat{i}-\hat{j}, 4 \hat{i}, 3…

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. Then the quadrilateral PQRS must be a
  1. parallelogram, which is neither a rhombus nor a rectangle.
  2. square.
  3. rectangle, but not a square.
  4. rhombus, but not a square.

Solution

$\mathrm{m}_{\mathrm{PQ}}=\frac{1}{6}, \mathrm{~m}_{\mathrm{SR}}=\frac{1}{6}, \mathrm{~m}_{\mathrm{RQ}}=-3, \mathrm{~m}_{\mathrm{SP}}=-3$
$\square \mathrm{PQRS}$ is a parallelogram. But neither $\mathrm{PR}=\mathrm{SQ}$ nor $\mathrm{PR}^{\prime} \perp \mathrm{SQ}$. $\therefore \quad$ Parallelogram, which is neither a rhombus nor a rectangle.

Asked in: MHT CET 2024 (15 May Shift 2)

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