Question

D is the mid – point of side BC of $\Delta$ ABC and E is the mid – point BD. If O is the mid – point of AE, then prove that as $\Delta BOE=\frac{1}{8}ar\left(\Delta ABC\right)$

Answer 1

Join A and D to get AD median
Median divides the triangle into 2 triangles of equal area
Therefore, $ar\left(ABD\right)=\frac{1}{2}ar\left(ABC\right)$
Now, join A and E to get AE median
Similarly we can prove that $ar\left(ABD\right)=\frac{1}{2}ar\left(ABC\right)$
$\Rightarrow ar\left(ABE\right)=\frac{1}{4}ABC\left[becausear\right(ABD)=\frac{1}{2}ar(ABC\left)\right]\cdots \left(1\right)$
Join B and O and we get BO median
$Now,ar\left(BOE\right)=\frac{1}{2}ar\left(ABE\right)$
$\Rightarrow ar\left(BOE\right)=\frac{1}{2}\times \frac{1}{4}ar\left(ABC\right)$
$\Rightarrow ar\left(BOE\right)=\frac{1}{8}ar\left(ABC\right)$