Question

In ΔABC, DE ∥ BC. If AD = 3 cm, DB = 2 cm and DE = 6 cm, then BC = ?


(a) 9 cm
(b) 10 cm
(c) 12 cm
(d) 18 cm

Answer 1

(b) 10 cm
In Δ ABC, DE ∥ BC. Then,
$\frac{AD}{DB}=\frac{AE}{EC}$ ( By Thales' theorem)
$\Rightarrow \frac{BD}{AD}=\frac{EC}{AE}$
$\Rightarrow \frac{BD}{AD}+1=\frac{EC}{AE}+1$
$\Rightarrow \frac{BD+AD}{AD}=\frac{EC+AE}{AE}$
$\Rightarrow \frac{AB}{AD}=\frac{AC}{AE}$ (Since BD + AD = AB and EC + AE = AC)
and $\angle A=\angle A$ (Common)
∴ $△ABC~△ADE$ (SAS similarity)
Thus,
$\frac{AB}{AD}=\frac{BC}{DE}$ [In similar triangles, corresponding sides are similar]
$\Rightarrow \frac{5}{3}=\frac{BC}{6}$
$\Rightarrow BC=\left(\frac{6\times 5}{3}\right)=10$ cm