In ∆ABC, a line segment DE is parallel to BC, with D on AB and E on AC. If the ratio of the area of △ADE to the area of the trapezoid DECB is 4 : 21, what is the ratio of AD to DB?

Correct option is A

Given :

In $$\triangle$$ABC , DE$$\parallel$$​BC with D on AB and E on AC.​
Area of $$\triangle$$​ADE : Area of trapezoid DECB = 4 : 21

Formula Used :

For similar triangles:
​$$\frac{\text{Area of } \triangle ADE}{\text{Area of } \triangle ABC}$$​​
=​$$\left(\frac{AD}{AB}\right)^2$$​​
Solution :

Let area of $$\triangle$$​ ADE = 4x 

Area of trapezoid DECB = 21x

So,
Area of $$\triangle$$​ ABC = 4x + 21x = 25x

Now,
​$$\frac{\text{Area of } \triangle ADE}{\text{Area of } \triangle ABC}$$​​

​$$=\frac{4x}{25x}$$​​

$$=\frac{4}{25}$$​​

Using area–side relation:
​$$\left(\frac{AD}{AB}\right)^2 = \frac{4}{25}$$​​

​$$\frac{AD}{AB} = \frac{2}{5}$$​​

So,
AB = AD + DB = 5 parts
AD = 2 parts, DB = 3 parts
AD : DB = 2 : 3