If $ABCD$ is parallelogram, $P$ is a point on side $BC$ and $DP$ when produced meets $AB$ produced at $L$, then prove that $\frac{DP}{PL}=\frac{DC}{BL}$.

Given: $ABCD$ is parallelogram, $P$ is a point on side $BC$ and $DP$ when produced meets $AB$ produced at $L$.

To do: To prove that $\frac{DP}{PL}=\frac{DC}{BL}$.

Solution:

As given A parallelogram $ABCD$ in which $P$ is a point on side $BC$ such that $DP$ produced meets $AB$ produced at $L$.

In $\vartriangle ALD$, we have

$BP||AD$

$\therefore \frac{LB}{BA}=\frac{LP}{PD}$

$\Rightarrow \frac{BL}{AB}=\frac{PL}{DP}$

$\Rightarrow \frac{BL}{DC}=\frac{PL}{DP}$     [$AB=DC$]

On taking reciprocal of both sides

$\Rightarrow \frac{DP}{PL}=\frac{DC}{BL}$

Hence proved.

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Updated on: 10-Oct-2022

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