Question

In the given figure, ABC is a triangle, DE is parallel to BC and $\frac{AD}{DB}=\frac{3}{2}$. [4 MARKS]


i) Determine the ratios $\frac{AD}{AB},\frac{DE}{BC}$

ii) Prove that $\Delta DEF$ is similar to $\Delta CBF$. Hence, find $\frac{EF}{FB}$

iii) What is the ratio of the areas of $\Delta DEF$ and $\Delta BFC$??

Answer 1

Applying theorems: 2 Marks
Calculation: 2 Marks

Given: $\Delta ABC,DE\parallel BC$ and $\frac{AD}{DB}=\frac{3}{2}$

In $\Delta ADE$ and $\Delta ABC$, $DE\parallel BC$
$\therefore \angle ADE=\angle ABC$ [Corresponding angles]
$\Rightarrow \angle A=\angle A$ [Common]
$\Rightarrow \Delta ADE\sim \Delta ABC$ [By AA similarity]

i) In $\Delta ADE$ and $\Delta ABC$,
$\frac{AD}{AB}=\frac{AD}{AD+DB}=\frac{3}{5}$
$\frac{DE}{BC}=\frac{AD}{AB}=\frac{3}{5}$

ii) In $\Delta DEF$ and $\Delta CBF$, $DE\parallel BC$
$\therefore \angle EDC=\angle DCB$ [Alternater angles]
$\Rightarrow \angle DEB=\angle EBC$ [Alterante angles]
$\Delta DEF\sim \Delta CBF$ [By AA similarity]
Hence, $\frac{EF}{FB}=\frac{DE}{BC}=\frac{3}{5}$

iii) $\frac{Areaof\Delta EFD}{Areaof\Delta BFC}=\frac{B{E}^2}{B{C}^2}=\frac{3^2}{5^2}=\frac{9}{25}$