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)


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