Question

In $△ABC,AD$ is the median to $BC$ and in $△PQR$, $PM$ is the median to $QR$. If $\frac{AB}{PQ}=\frac{BC}{QR}=\frac{AD}{PM}$. Prove that $△ABC\sim △PQR$.

Answer 1

In $\Delta ABD$ and $\Delta PQM$

$\frac{AB}{PQ}=\frac{AD}{PM}=\frac{BC}{QR}=\frac{BC}{2}\times \frac{2}{QR}=\frac{BD}{QM}$

Hence all $3$ sides are in proportion.

$\Delta ABD\sim \Delta PQM\left(SSS\right)$

So, $\angle ABD=\angle PQM$

Now in $\Delta ABC$ & $\Delta PQM$

$\frac{AB}{PQ}=\frac{BC}{QR}$ and $\angle ABD=\angle PQM$ (SAS)

Hence $\Delta ABD$ and $\Delta PQM$ (SAS)

Hence, proved.