In a ΔABC, DE ∥ BC, where D is a point on AB and E is a point on AC. If DE divides the area of ΔABC into two equal parts, then DB ∶ AB is equal to :

Question

Question

This question was previously asked in

SSC CGL 2023 Tier-I Official Paper (Held On: 20 Jul 2023 Shift 4)

Answer 1

Concept used

The ratio of the areas of two similar triangles is equal to the ratio of the squares of the corresponding sides & medians.

Calculation

As DE divides the area of ΔABC into two equal parts.

i.e. ar(ΔADE) = ar(DEBC)

⇒ ar(ΔADE) = ar(ΔABC)/2

⇒ ar(ΔABC)/ar(ΔADE) = 2

Now,

ΔABC ∼ ΔADE (By AAA similarity)

So,

ar(ΔABC)/ar(ΔADE) = (AB/AD)²

⇒ 2 = AB²/AD²

⇒ AB2/AD2 = 2

⇒ AB = √2AD

⇒ AB = √2(AB - DB)

⇒ √2AB - AB = √2DB

⇒ AB(√2 - 1) = √2DB

∴ DB/AB = (√2 - 1)/√2

The answer is (√2 - 1):√2

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