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

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

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Solution.

Given: O is any point on the diagonal PR of a parallelogram PQRS
Construction: Join QS
Let diagonals PR and QS intersect each other at T.

We know that diagonals of a parallelogram bisect each other. 
\ T is the mid-point of QS and PR 
Since a median of a triangle divides it into two triangle of equal area. 
\ In \triangle PQS, PT is the median of side QS
\Rightarrow ar(\triangle PTS)=ar(\triangle PTQ)                    …(i)
In \triangle SOQ, OT is the median of side QS
\Rightarrow ar(\triangle STO)=(\triangle QTO)                       …(ii)
Adding (i) and (ii), we have
ar(\triangle PTS)+ar(\triangle STO)=ar (\triangle PTQ)+(\triangle QTO)
\Rightarrow ar(\triangle PSO)=ar(\triangle PQO)
Hence proved

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