Question

In figure, $DE\parallel BC$ and $AD:DB=5:4$ . Find $\frac{\text{Area}\left(ADE\right)}{\text{Area}\left(ABC\right)}$.

Answer 1

$\because DE||BC$

$AD:DB=5:4$

$\Delta ADM\sim \Delta ABN$

$\therefore \frac{AD}{AB}=\frac{AM}{AN}$

$\frac{5}{9}=\frac{AM}{AN}$ _____ (i)

$\because \frac{AD}{AB}=\frac{DE}{BC}$

$\frac{5}{9}=\frac{DE}{BC}$ ______ (ii)

$ar\left(\Delta ADE\right)=\frac{1}{2}\times DE\times AM$ ______ (iii)

$ar\left(\Delta ABC\right)=\frac{1}{2}\times BC\times AN$ _______ (iv)

(iii) & (iv)

$\frac{ar\left(\Delta ADE\right)}{ar\left(\Delta ABC\right)}=\frac{DE}{BC}\times \frac{AM}{AN}$

$=\frac{5}{9}\times \frac{5}{9}$

$\frac{ar\left(\Delta ADE\right)}{ar\left(\Delta ABC\right)}=\frac{25}{81}$