Question

If two side and a median bisecting the third side of a triangle are respectively proportional to the corresponding sides and the median of another triangle, then two triangles are similar.

• A. True
• B. False

Answer 1

The correct option is A True

Given :

$△ABC$ and $△PQR$ in which $AL$ and $PM$ are themedians such that,

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

To prove :

$△ABC\sim △PQR$

Proof :

$\frac{AB}{PQ}=\frac{BC}{QR}=\frac{AL}{PM}$ [ Given ]

As we know that a median bisects a side into two equal parts.

$\Rightarrow$ $BL=LC$ and $QM=MR$

$\Rightarrow$ $BC=2BL$ and $QR=2QM$

$\therefore$ $\frac{AB}{PQ}=\frac{2BL}{2QM}=\frac{AL}{PM}$

$\Rightarrow$ $\frac{AB}{PQ}=\frac{BL}{QM}=\frac{AL}{PM}$

$\Rightarrow$ $△ABC\sim △PQM$ [ By $SSS$ similarity test ]

$\Rightarrow$ $\angle B=\angle Q$ [ C.P.S.T ]

Now, In $△ABC\sim △PQR,$ we have

$\Rightarrow$ $\frac{AB}{PQ}=\frac{BC}{QR}$ and $\angle B=\angle Q$

$\therefore$ $△ABC\sim △PQR$ [ $SAS$ similarity test ]