Question

In triangle ABC, D and E are the points on the sides AB and AC. DE||BC, BD=5 cm and DE:BC=2:3. Find AD and the ratio of are of triangle ADE to the quadrilateral BCED.

Answer 1

$DE||BC$

$\Rightarrow △ADE\sim △ABC$

$\Rightarrow AD/AB=DE/BC=AE/AC$

$DE/BC=2/3$

$\Rightarrow AD/AB=2/3$

$\Rightarrow AD/(AD+BD)=2/3$

$\Rightarrow AD/(AD+2)=2/3$

$\Rightarrow 3AD=2AD+4$

$\Rightarrow AD=4$

And

Ratio of Area of Similar triangle $=$ (Ratio of sides of similar triangle)${}^2$

$\Rightarrow$ Area of $△$ ADE / Area of $△ABC=(2/3{)}^2$

$\Rightarrow$ Area of $△ADE$ / Area of $△ABC=4/9$

$\Rightarrow$ Area of $△ABC=9\ast$ Area of $△ADE/4$

Area of quadrilateral bced $=$ Area of $\Delta ABC-$ Area of $\Delta ADE$

$\Rightarrow$ Area of quadrilateral becd $=9\ast$ Area of $△ADE/4-$ Area of $△ADE$

$\Rightarrow$ Area of quadrilateral bced $=5\ast$ Area of $△ADE/4$

$\Rightarrow$ area of $△ADE/$Area of quadrilateral bced $=4/5$