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Question:

In given figure, ABC and DBC are two triangles on the same base BC. If AD intersects BC at O, show that: area(ABC)/area(DBC) = AO/DO

Solution:

Construction:DrawAMperpendicular toBCandDNperpendicular toBC.Now,In△AMOand△DNO∠AOM=∠DON...opp.angles∠AMO=∠DNO=90o∴By AA Criterion of Similarity,△AMO∼△DNO∴AMDN=AODO(Corresponding Sides of Similar Triangles). (1)Now,A(△ABC)A(△BDC)=12×BC×AM12×BC×DN∴A(△ABC)A(△BDC)=AODO(From 1)[henceproved]