Question

in a triangle ABC D is the midpoint of BC and e is the midpoint of BD and O is the m.p AE then prove that area of triangle BOE= 1/8 area of ABC.

Answer 1

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)

......