Question

In the given figure, ΔABC and. ΔDBC are on the same base BC. AD and BC intersect at O. Prove that ar(ΔABC)/ ar(ΔDBC) = AO / DO

Answer 1

attach the figure please

Answer 2

Since ΔABC and ΔDBC are one same base, Therefore ratio between their areas will be as ratio of their heights. Let us draw two perpendiculars AP and DM on line BC.

In ΔAPO and ΔDMO,

∠APO = ∠DMO (Each is 90°)

∠AOP = ∠DOM (vertically opposite angles)

∠OAP = ∠ODM (remaining angle)

Therefore ΔAPO congruent to ΔDMO (By AAA rule)

Therefore, AP/DM=AO/DO

ar(ΔABC)/ ar(ΔDBC) = AO / DO

Hence Proved