Q.3 (B)(1) Show that points P(1, –2), Q(5, 2), R(3, –1), S(–1, –5) are the vertices of a parallelogram.

Answer: The diagonals PR and QS have the same midpoint (2, –3/2), so PQRS is a parallelogram.

Midpoint of diagonal PR = \(\left(\dfrac{1 + 3}{2}, \dfrac{-2 - 1}{2}\right) = \left(2, -\dfrac32\right)\).

Midpoint of diagonal QS = \(\left(\dfrac{5 - 1}{2}, \dfrac{2 - 5}{2}\right) = \left(2, -\dfrac32\right)\).

The diagonals bisect each other, so □PQRS is a parallelogram.

(Check with slopes: PQ = \(\tfrac{4}{4} = 1\), SR = \(\tfrac{4}{4} = 1\); QR = \(\tfrac{-3}{-2} = \tfrac32\), PS = \(\tfrac{-3}{-2} = \tfrac32\). Both pairs of opposite sides are parallel.)