Question

Side $AB$ and $BC$ and median $AD$ of a triangle $ABC$ are respectively proportional to sides $PQ$ and $QR$ and median $PM$ of $PQR$. Show that $ABC\sim PQR$ .

Answer 1

Given :

A $△$ ABC where AD is the median, $△$ PQR where PM is the median.

And, $\frac{AB}{PQ}=\frac{BC}{QR}=\frac{AD}{PM}$

To prove :

$△ABC\sim △PQR$

Proof :

Since AD is the median,

$BD=CD=\frac{1}{2}BC$

Similarly, PM is the median,

$QM=RM=\frac{1}{2}QR$

Given that

$\frac{AB}{PQ}=\frac{BC}{QR}=\frac{AD}{PM}$

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

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

Since all 3 sides are proportional

$△ABD\sim △PQM$ [By SSS similarity]

Hence, $\angle B=\angle Q$

In $△ABC$ and $△PQR$

$\angle B=\angle Q$

$\frac{AB}{PQ}=\frac{BC}{QR}$

Hence, by SAS similarity,

$△ABC\sim △PQR$