In the given figure, $PQRS$ is a parallelogram.....

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In the given figure, $PQRS$ is a parallelogram. $PQ$ is produced to $L$ so that $QL=PQ$. The line $SL$ cuts $QR$ at $O$. Prove that: $\triangle PQS=2\triangle ROL$.

Attempt

$$PL=LQ$$ $$ Q \ \text{is the mid point of} \ PC$$ $$\triangle POQ=\triangle QOL$$

Now what should I do next?

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1 Answer

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Construction:-Join $SQ$.

Proof:-

$\color{red}{\triangle PSQ=\triangle SQL}$(as Q is the mid point)......(1)

Also,$\color{blue}{\triangle SQL=\triangle RQL}$(as between same parallels and same base).....(2)

Also,note that $\color{orange}{\triangle SOR\cong\triangle QOL}\space(AAS)$

So,$O$ is the mid point of $RQ$(cpct)

So,$\color{brown}{\triangle ROL=\triangle QOL}$.

So,$\color{magenta}{\triangle RQL=2\triangle ROL}$.....(3)

So,comparing (1),(2),(3) we get,$\color{green}{\triangle PQS=2\triangle ROL}$.

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