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If AD and PM are medians of triangles ABC and PQR, respectively where $\Delta ABC \sim \Delta PQR$, prove that $\frac{AB}{PQ} = \frac{AD}{PM}$
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Sol.
AD and AM are medians of $\Delta ABC$ and $\Delta PQR$ respectively.
$\Delta ABC \sim \Delta PQR$
$\therefore \frac{AB}{PQ} = \frac{BC}{QR}$
$\frac{AB}{PQ} = \frac{2BD}{2QM}$
$\frac{AB}{PQ} = \frac{BD}{QM}$
Also $\angle B = \angle Q$ ($\Delta ABC \sim \Delta PQR$)
$\Rightarrow \Delta ABD \sim \Delta PQM$ (SAS similarly)
$\Rightarrow \frac{AB}{PQ} = \frac{AD}{PM}$_a_1.png)
AD and AM are medians of $\Delta ABC$ and $\Delta PQR$ respectively.
$\Delta ABC \sim \Delta PQR$
$\therefore \frac{AB}{PQ} = \frac{BC}{QR}$
$\frac{AB}{PQ} = \frac{2BD}{2QM}$
$\frac{AB}{PQ} = \frac{BD}{QM}$
Also $\angle B = \angle Q$ ($\Delta ABC \sim \Delta PQR$)
$\Rightarrow \Delta ABD \sim \Delta PQM$ (SAS similarly)
$\Rightarrow \frac{AB}{PQ} = \frac{AD}{PM}$