Question

In given figure, $P$ and $Q$ are points on the sides $AB$ and $AC$ respectively of $△ABC$ such that $AP=3.5cm,PB=7cm,AQ=3cm$ and $QC=6cm$. If $PQ=4.5cm$, find $BC$.

Answer 1

We have

$\left(\frac{AQ}{AC}\right)=\frac{3}{9}=\frac{1}{3}$

$\left(\frac{AP}{AB}\right)=\frac{3.5}{10.5}=\frac{1}{3}$

In $\Delta APQ$ & $\Delta ABC$

$\left(\frac{AQ}{AC}\right)=\left(\frac{AP}{AB}\right)$

& $\angle PAQ=\angle BAC$

Thus $\Delta APQ\sim \Delta ABC$ by SAS criterion.

Hence $\left(\frac{AQ}{AC}\right)=\left(\frac{PQ}{BC}\right)$ (by cpst)

$\Rightarrow \frac{1}{3}=\left(\frac{4.5}{BC}\right)$

$\Rightarrow BC=13.5$cm.