Question

D is the mid-point of side BC of $\Delta ABC$ and E is the mid point of BD. If O is the midpoint of AE.Then Area of ($\Delta BOE$) .

• A. $\frac{1}{8}Area\left(\Delta ABC\right)$
• B. $\frac{1}{6}Area\left(\Delta ABC\right)$
• C. $\frac{1}{4}Area\left(\Delta ABC\right)$
• D. $\frac{1}{3}Area\left(\Delta ABC\right)$

Answer 1

The correct option is A $\frac{1}{8}Area\left(\Delta ABC\right)$


Since AD is the median,
$\therefore 2ar\left(\Delta ABD\right)=ar\left(\Delta ABC\right)$ ...(i)
$Similarly,2ar\left(\Delta ABE\right)=ar\left(\Delta ABD\right)...\left(ii\right)$
$And2ar\left(\Delta BOE\right)=ar\left(\Delta ABE\right)...\left(iii\right)$
$Now,ar\left(\Delta ABC\right)=2ar\left(\Delta ABD\right)$
$=2\left[2ar\right(\Delta ABE\left)\right]$ from (ii)
$=4\left[2ar\right(\Delta BOE\left)\right]$ from (iii)
$=8ar\left(\Delta BOE\right)$
$\therefore \frac{1}{8}ar\left(\Delta ABC\right)=ar\left(\Delta BOE\right)$