Question

In the given figure, If PQ||BC and PR||CD. Prove that

(i) $\frac{AR}{AD}=\frac{AQ}{AB}$

(ii) $\frac{QB}{AQ}=\frac{DR}{AR}$

Answer 1

In $△ABC$, we have

$PQ||BC$

Therefore, by basic proportionality theorem, we have

$\frac{AQ}{AB}=\frac{AP}{AC}$........(i)

In $△ACD$, we have

$PR||CD$

Therefore, by basic proportionality theorem, we have

$\frac{AP}{AC}=\frac{AR}{AD}$

From (i) and (ii), we obtain that

$\frac{AQ}{AB}=\frac{AR}{AD}$ or $\frac{AR}{AD}=\frac{AQ}{AB}$ $\left[Henceproved\right]$

$\Rightarrow$ $\frac{AB}{AQ}=\frac{AD}{AR}$

$\Rightarrow$ $\frac{AQ+QB}{AQ}=\frac{AR+RD}{AR}$

$\Rightarrow$ $1+\frac{QB}{AQ}$ $=1+\frac{RD}{AR}$

$\Rightarrow \frac{QB}{AQ}=\frac{DR}{AR}$ $\left[Henceproved\right]$