In the adjoining figure, ABCD is a ‖ gm and O is a point on the diagonal AC. Prove that ar(∆AOB) = ar(∆AOD).
Given: ABCD is a ‖ gm and O is a point on the diagonal AC.
Construction: Drop perpendiculars DM and BN onto diagonal AC.
Here,
DM = BN (perpendiculars drawn from opposite vertices of a ||gm to the diagonal are equal)
Now,
Area (ΔAOB) = 1/2 × base × height = 1/2 × AO × BN –1
Area (ΔAOD) = 1/2 × base × height = 1/2 × AO × DM –2
From –1 and –2
Area (∆AOB) = Area (∆AOD) (∵ BN = DM)