Asked in: March 2025 board paperShow that points P(1, –2), Q(5, 2), R(3, –1), S(–1, –5) are the vertices of a parallelogram.

Answer: Both pairs of opposite sides of □PQRS are parallel, so it is a parallelogram.

Slope of PQ \(= \dfrac{2 - (-2)}{5 - 1} = \dfrac{4}{4} = 1\)

Slope of QR \(= \dfrac{-1 - 2}{3 - 5} = \dfrac{-3}{-2} = \dfrac{3}{2}\)

Slope of RS \(= \dfrac{-5 - (-1)}{-1 - 3} = \dfrac{-4}{-4} = 1\)

Slope of SP \(= \dfrac{-2 - (-5)}{1 - (-1)} = \dfrac{3}{2}\)

Slope of PQ = slope of RS, so side PQ \(\parallel\) side RS. Slope of QR = slope of SP, so side QR \(\parallel\) side SP.

Both pairs of opposite sides are parallel. \(\therefore\) □PQRS is a parallelogram, i.e. P, Q, R, S are its vertices.