Question

In fig., $$DE || BC, \, AD = 1 \, cm$$ and $$BD = 2 cm$$. What is the ratio of the $$ar (\Delta ABC)$$ to the $$ar (\Delta ADE)$$ ?

Answer 1

Consider the $$\triangle ABC,\triangle ADE$$

$$DE||BC $$

$$ \implies \angle ADE=\angle ABC $$

$$ \angle AED=\angle ACB $$

$$ \angle DAE=\angle BAC$$

So, the triangles are similar

$$ \implies \dfrac{ar\left(\Delta ABC\right)}{ar\left(\Delta ADE\right)}=\left(\dfrac{AB}{AD}\right)^2$$ $$\left(AB=AD+DB=1+2=3 \text{cm}\right)$$

$$\implies \left(\dfrac{3}{1}\right)^2 =9$$

So the Ratio is $$9:1$$