Question

If D is a point on the side AB of ∆ABC such that AD : DB = 3.2 and E is a point on BC such that DE || AC. Find the ratio of areas of ∆ABC and ∆BDE.

Answer 1

Given: In ΔABC, D is a point on side AB such that AD : DB= 3 : 2. E is a point on side BC such that DE || AC.

To find:

In ΔABC,

According to converse of basic proportionality theorem if a line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side.

Hence DE || AC

In ΔBDE and ΔABC,

$\angle \mathrm{BDE}=\angle \mathrm{A}\left(\mathrm{Corresponding}\mathrm{angles}\right)$

$\angle \mathrm{DBE}=\angle \mathrm{ABC}\left(\mathrm{Common}\right)$

So, $∆\mathrm{BDE}~∆\mathrm{ABC}$ (AA Similarity)

We know that the ratio of areas of two similar triangles is equal to the ratio of squares of their corresponding sides.

Let AD = 2x and BD = 3x.

Hence