Question

In Figure, ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, show that `(ar(ABC))/(ar(DBC)) = (AO)/(DO)`

Answer 1

Let us draw two perpendiculars AP and DM on line BC.

We know that area of a triangle = `1/2 xx "Base" xx "Height"`

`:.(ar(triangleABC))/(ar(triangleDBC)) =  (1/2 BC xx AP)/(1/2BC xx DM) = (AP)/(DM)`

In ΔAPO and ΔDMO,

∠APO = ∠DMO (Each = 90°)

∠AOP = ∠DOM (Vertically opposite angles)

∴ ΔAPO ∼ ΔDMO (By AA similarity criterion)

`:. (AP)/(DM) = (AO)/(DO)`

`=> (ar(triangleABC))/(ar(triangleDBC))=(AO)/(DO)`