O is any point on the diagonal PR of a parallelogram PQRS (figure). Prove that `ar (DeltaPSO) = ar (DeltaPQO)`.

Explanation

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

To prove `ar (DeltaPSO) = ar (DeltaPQO)`
Construction Join SQ which intersects 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 `DeltaQPS` and we know that, a median of a triangle divides it into two triangles of equal area.
`therefore" "` `ar (DeltaBPQ) = ar (DeltaBPS)" "` ...(i)
Also, OB is the median of `DeltaOSQ`.
`therefore" "` `ar (DeltaOBQ) = ar (DeltaOBS)" "` ...(ii)
On adding Eqs. (i) and (ii), we get
`ar (DeltaBPQ) + ar (DeltaOBQ) = ar (DeltaBPS) + ar (DeltaOBS)`
`rArr" "` `ar (DeltaPQO) = ar (DeltaPSO)" "` Hence proved.