Question

D is midpoint of side BC of $△ABC$ and E is the midpoint of BD.If O is the midpoint of AE,prove that $ar\left(△BOE\right)=\frac{1}{8}ar\left(△ABC\right)$

Answer 1

ANSWER:
D is the midpoint of side BC of ∆ABC.
⇒AD is the median of ∆ABC.
⇒ar△ABD=ar△ACD=1/2ar△ABC
E is the midpoint of BD of ∆ABD,
⇒AE is the median of ∆ABD
⇒ar△ABE=ar△AED=1/2ar△ABD=1/4ar△ABC
Also, O is the midpoint of AE,
⇒BO is the median of ∆ABE,
⇒ar△ABO=ar△BOE=1/2ar△ABE=1/4ar△ABD=1/8ar△ABC
Thus, ar(∆ BOE) = 1/8ar(∆ ABC)