Question

In Fig., 4.179, ΔABC and ΔDBC are on the same base BC. If AD and BC intersect at O,

prove that  area (ΔABC)/area(Δ DBC)=AO/DO

Answer 1

We know that area of a triangle = 1/2 x Base x height

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 ~ ΔDMO (By AAA rule)