If AD and PM are medians of triangles ABC and PQR respectively, where$∆ABC \sim ∆PQR$. Prove that $\frac{AB}{PQ}=\frac{AD}{PM}$∙

AcademicMathematicsNCERTClass 10


$AD$ and $PM$ are medians of $\vartriangle ABC$ and $\vartriangle PQR$ respectively where, $\vartriangle ABC\sim \vartriangle PQR$. 

To do:

We have to prove that $\frac{AB}{PQ}=\frac{AD}{PM}$.


It is given that $\vartriangle ABC \sim   \vartriangle PQR$

We know that,

The corresponding sides of similar triangles are always proportional.

This implies,


$\angle A = \angle P,\ \angle B = \angle Q,\ \angle C = \angle R$............(ii)

$AD$ and $PM$ are medians.

This implies,

They divide their opposite sides $BC$ and $QR$ respectively.


$BD=\frac{BC}{2}$ $QM=\frac{QR}{2}$.......(iii)

From equations (i) and (iii), we get,


In $\vartriangle ABD$ and $\vartriangle PQM$,

$\angle B = \angle Q$         [From (ii)]

$\frac{AB}{PQ}=\frac{BD}{QM}$     [From (iv)]

Therefore, by SAS similarity,

$\vartriangle ABD \sim   \vartriangle PQM$

$\Rightarrow \frac{AB}{PQ}=\frac{BD}{QM}=\frac{AD}{PM}$

Hence proved.

Updated on 10-Oct-2022 13:21:19