In $$\triangle$$ABC, D and E are the points on sides AB and AC, respectively, such that DE $$\parallel$$ BC. If DE : BC is 3 : 5, then (Area of $$\triangle$$ADE) : (Area of quadrilateral DECB) is:

In $$\triangle$$ABC, D and E are the points on sides AB and AC, respectively, such that DE $$\parallel$$ BC.

In $$\triangle$$ABC and $$\triangle$$ADE

$$\angle A\ $$ is common among both the triangles.

$$\angle ABC\ $$ = $$\angle ADE\ $$

$$\angle ACB\ $$ = $$\angle AED\ $$

As per AAA property, $$\triangle$$ABC $$\simeq$$ $$\triangle$$ADE [both are similar triangles.]

If DE : BC is 3 : 5.

Let's assume DE = 3y and BC = 5y.

$$\frac{Area\ of\ \triangle\ ADE}{Area\ of\ \triangle\ ABC}\ =\ \left(\frac{DE}{BC}\right)^2$$

$$\frac{Area\ of\ \triangle\ ADE}{Area\ of\ \triangle\ ABC}\ =\ \left(\frac{3y}{5y}\right)^2$$

$$\frac{Area\ of\ \triangle\ ADE}{Area\ of\ \triangle\ ABC}\ =\ \frac{9}{25}$$
Area of quadrilateral DECB = 25-9 = 16

(Area of $$\triangle$$ADE) : (Area of quadrilateral DECB) = 9 : 16