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Question

D is the mid – point of side BC of Δ ABC and E is the mid – point BD. If O is the mid – point of AE, then prove that as ΔBOE=18ar(ΔABC)


Solution

Join A and D to get AD median
Median divides the triangle into 2 triangles of equal area
Therefore, ar(ABD)=12ar(ABC)
Now, join A and E to get AE median
Similarly we can prove that ar(ABD)=12ar(ABC)
ar(ABE)=14ABC [because ar(ABD)=12ar(ABC)](1)
Join B and O and we get BO median
Now, ar(BOE)=12 ar(ABE)
ar(BOE)=12×14ar(ABC)
ar(BOE)=18ar(ABC)

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