The base BC of ∆ABC is divided at D such BD = DC. Prove that ar(∆ABD) = x ar(∆ABC).

Given: A ∆ABC with a point D on BC such that BD = DC

To prove: area(∆ABD) = x area(∆ABC)

Construction: Drop a perpendicular onto BC

Proof: area(∆ABC) = x BC x AE ---------------(1)

and, area(∆ABD) = x BD x AE ----------------- (2)

given that BD = DC ------------------ (3)

so, BC = BD + DC = BD + 2BD = 3BD [from 2]

∴ BD = (BC)

Sub BD in (1), we get

area(∆ABD) = x ((BC) X AE)

area(∆ABD) = x ( BC X AE)

area(∆ABD) = x area(∆ABC) [from 1]

Hence proved