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Question 6
O is any point on the diagonal PR of a parallelogram PQRS (figure). Prove that ar (Δ PSO) = ar (Δ PQO).

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Solution

Construction: Join SQ which intersects PR at B .


Proof:
We know that diagonals of a parallelogram bisect each other. So, B is the midpoint of SQ.
Here, PB is a median of Δ QPS and we know that, a median of a triangle divides it into two triangles of equal area.
ar(Δ BPQ) = ar (ΔBPS) ...(i)
ar(Δ OBQ) = ar (ΔOBS) ....(ii)
On adding Eqs. (i) and (ii), we get,
ar(ΔBPQ)+ar(ΔOBQ)=ar(ΔBPS)+ar(ΔOBS)
ar(ΔPQO)=ar(ΔPSO)


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