Question

In triangle △ABC, AD : DB = 2 : 3, DE is parallel to BC.

Assertion (A): \[\frac{\mathrm{DE}}{\mathrm{BC}}=\frac{\mathrm{AD}}{\mathrm{BD}}=\frac{2}{3}\].

Reason (R): \[\frac{\mathrm{DE}}{\mathrm{BC}}=\frac{\mathrm{AD}}{\mathrm{AB}}=\frac{2}{5}\]

Options

• A is true, R is false.

• A is false, R is true.

• Both A and R are true and R is the correct reason for A.

• Both A and R are true and R is the incorrect reason for A.

Answer 1

A is false, R is true.

Explanation:

Given,

AD : DB = 2 : 3

Let AD = 2x and DB = 3x.

From figure,

AB = AD + DB = 2x + 3x = 5x.

In Δ ADE and Δ ABC

⇒ ∠DAE = ∠BAC (Common angle)

⇒ ∠ADE = ∠ABC (Corresponding angles are equal)

⇒ ∠AED = ∠ACB (Corresponding angles are equal)

∴ ΔADE ∼ ΔABC (By AAA postulate)

We know that,

In similar triangles, corresponding sides are proportional (or in the same ratio).

∴ `(DE)/(BC) = (AD)/(AB)`

⇒ `(DE)/(BC) = (2x)/(5x) = 2/5`

∴ Assertion (A) is false, reason (R) is true.