And Are Points On Sides And Respectively Of Оґabc Such That Bc And Divides Оґabc Into Two Parts

D and E are points on the sides AB and AD respectively of a ΔABC such that DE ∥ BC and divides ΔABC into two parts, equal in area….

Answers:

We have

area(ΔADE) = area (trapezium BCED)

⇒ area (ΔADE) + area(ΔADE)

= area(trapezium BCED) + area(ΔADE)

⇒ 2 area(ΔADE) = area(ΔABC)   ….(i)

In ΔADE  and ΔABC, we have

∠ADE = ∠B    [∵  DE ∥ BC]

∴   ∠AED = ∠C (corresponding angles)]

and  ∠A = ∠A,     [Common]

∴  ΔADE ~ ΔABC

⇒ area(ΔADE)/area(ΔABC) = AD²/AB²

⇒ area(ΔADE)/2 area(ΔADE) = AD²/AB²

⇒ ½ = (AD/AB)²

⇒ AD/AB = 1/√2

⇒ AB = √2 AD

⇒AB = √2(AB – BD)

⇒ (√2 – 1)AB =  √2BD

⇒ BD/AB = (√2 – 1)/√2

= [(2 – √2)/2]