Question

If $\Delta ABC\sim \Delta PQR$ and AD and PM are corresponding medians of the two triangles, then

• A. $\frac{AB}{PM}=\frac{AD}{PQ}$
• B. $\frac{AB}{PR}=\frac{AD}{PM}$
• C. $\frac{AB}{PQ}=\frac{AD}{PM}$
• D. $\frac{AC}{PQ}=\frac{AD}{PM}$

Answer 1

The correct option is C

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

Given, $\Delta ABC\sim \Delta PQR$

AD and PM are the medians of $\Delta ABC$ and $\Delta PQR$ respectively.

$\Delta ABC\sim \Delta PQR$

$\therefore \angle B=\angle Q$ and
$\frac{AB}{PQ}=\frac{BC}{QR}=\frac{AC}{PR}$
Consider $\frac{AB}{PQ}=\frac{BC}{QR}\frac{AB}{PQ}=\frac{\frac{1}{2}BC}{\frac{1}{2}QR}=\frac{BD}{QM}$

$(\because$ D and M are mid-points of BC and QR)

Now in $\Delta ABD$ and $\Delta PQM$

$\frac{AB}{PQ}=\frac{BD}{QM}$ (Proved)

$\angle B=\angle Q$ (given)

$\therefore \Delta ABD\sim \Delta PQM$ (SAS axiom)

$\therefore \frac{AB}{PQ}=\frac{AD}{PM}$

(corresponding sides of similar triangles are proportional)