Question

D and E are points on the sides AB and AC respectively of a Δ ABC such that DE | | BC and divides Δ ABC into two parts, equal in area. Find `"BD"/"AB"`.

Answer 1

We have
area (Δ ADE) = area (trapezium BCED)
⇒ area (Δ ADE) + area (Δ ADE)
= area (trapezium BCED) + area (Δ ADE)
⇒ 2 area (Δ ADE) = area (Δ ABC)   ...(i)
In Δ ADE and Δ ABC, we have
∠ADE = ∠B,      ...[∵ DE | | BC]
∴ ∠AED = ∠C   ...(corresponding angles)]
and ∠A = ∠A,    ...[Common]

∴ Δ ADE ∼ Δ ABC
⇒ `"area (Δ ADE)"/"area (Δ ABC)" = "AD"^2/"AB"^2`
⇒ `"area (Δ ADE)"/(2"area (Δ ADE)") = "AD"^2/"AB"^2`
⇒ `(1)/(2) = ("AD"/"AB")^2`
⇒ `"AD"/"AB" = (1)/sqrt(2)`
⇒ AB = `sqrt(2)"AD"`
⇒ AB = `sqrt(2)("AB - BD")`
⇒ `(sqrt(2) - 1)"AB" = sqrt(2)"BD"`
⇒ `"BD"/"AB" = (sqrt(2) - 1)/(sqrt(2)`
= `(2 - sqrt(2)/(2))`.