Question

In a $\Delta ABC$, $D$ and $E$ are points on the sides $AB$ and $AC$ respectively such that $DE||BC$.
If $\frac{AD}{BD}=\frac{4}{5}$ and $EC=2.5cm$, find $AE$.

Answer 1

$\begin{array}{c}\frac{AD}{BD}=\frac{4}{5}EC=2.5cm.\\ In\Delta ABC\&\Delta ADC\\ \angle A=\angle A\&\angle ADE=\angle ABC\\ \therefore \Delta ABC\sim \Delta ADE\\ \frac{AB}{AD}=\frac{AC}{AE}\\ \Rightarrow \frac{AB}{AD}-1=\frac{AC}{AE}-1\\ \frac{AB-AD}{AD}=\frac{AC-AE}{AE}\\ \Rightarrow \frac{DB}{AD}=\frac{EC}{AE}\\ \frac{5}{4}=\frac{2.5}{AE}\\ \therefore AE=\frac{2.5}{5}\times 4=2cm.\end{array}$