Question

In a $△RST$, line $l\parallel$side $ST$ and line $l$ intersects side $RS$ and side $RT$ in points $P$ and $Q$ respectively, $R-P-S$ and $R-Q-T$, then prove that $\frac{RP}{PS}=\frac{RQ}{QT}$.

Answer 1

As $l||ST$ so $PQ||ST$

Now consider $△PRQ,△SRT$

$\angle PRQ=\angle SRT$ [common angle]

$\angle RPQ=\angle RST$ [corresponding angles]

$\angle RQT=\angle RTS$ [corresponding angles]

$△PRQ,△SRT$ are similar

$\frac{PR}{SR}=\frac{RQ}{RT}=\frac{PQ}{ST}$

$\Rightarrow \frac{SR-PR}{SR}=\frac{RT-RQ}{RT}$

$\Rightarrow \frac{PS}{SR}=\frac{QT}{RT}$

$\Rightarrow \frac{PS}{QT}=\frac{SR}{RT}$$......\left(1\right)$

Again $\Rightarrow \frac{PR}{RQ}=\frac{SR}{RT}...\left(2\right)$

Using the two equations

$\Rightarrow \frac{PS}{QT}=\frac{PR}{RQ}$

$\Rightarrow \frac{RP}{PS}=\frac{RQ}{QT}$

hence proved.