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Question

D is the mid point of side BC of triangle ABC and E is the mid point of BD. If O is the mid point of AE, then area of triangle BOE = 1/8 area of triangleABC


Solution

join A and D to get AD median

Median divides the triangle into 2 triangles of equal area

therefore, ar(ABD)=1/2 ar(ABC)

now, join A and E  to get AE median 

Similarly we can prove that ar(ABE) = 1/2 ABD

==> ar(ABE)= 1/4 ABC  [ because ar(ABD)= 1/2 ar(ABC) ] ---------1)

Join B and O and we get BO median

Now, ar(BOE)= 1/2 ar(ABE)

==>ar(BOE)= 1/2 x  1/4 ar(ABC)

==>ar(BOE)= 1/8 ar(ABC)

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