DEBC and AE:EC=3:4 are true for the given figure. What will the proportion between ADE and ABC be?

In $∆\mathrm{ADE}$ and $∆\mathrm{ABC}$

$\angle \mathrm{ADE}=\angle \mathrm{ABC}$ (Corresponding angles)

$\angle \mathrm{AED}=\angle \mathrm{ACB}$ (Corresponding angles)

$∆\mathrm{ADE}~∆\mathrm{ABC}$ (AA similarity)

We are aware that the ratio of the squares of the corresponding sides equals the ratio of the areas of similar triangles. The theorem states that

$\frac{\text{ area of }∆ADE}{\text{ area of }∆ABC}={\left(\frac{AE}{AC}\right)}^2$

 $\frac{\text{ area of triangle }ADE}{\text{ area of triangle }ABC}={\left(\frac{3x}{7x}\right)}^2$

 $\frac{\text{ area of }∆ADE}{\text{ area of }∆ABC}=\frac{9x^2}{49x^2}=\frac{9}{49}$