Question

O is any point on the diagonal PR of a parallelogram PQRS (Figure). Prove that ar (PSO) = ar (PQO).

Answer 1

Given: In a parallelogram PQRS, O is any point on the diagonal PR.

To prove: ar (ΔPSO) = ar (ΔPQO)

Construction: Join SQ which intersect PR at B.

Proof: We know that, diagonals of a parallelogram bisect each other, so B is the mid-point of SQ.

Here, PB is a median of ΔQPS and we know that, a median of a triangle divides it into two triangles of equal area.

∴ ar (ΔBPQ) = ar (ΔBPS)  ...(i)

Also, OB is the median of ΔOSQ.

∴ ar (ΔOBQ) = ar (ΔOBS)  ...(ii)

On adding equations (i) and (ii), we get

ar (ΔBPQ) + ar (ΔOBQ) = ar (ΔBPS) + ar (ΔOBS)

⇒ ar (ΔPQO) = ar (ΔPSO)

Hence proved.