Question

D is the mid-point of side BC of ΔABC and E is the mid-point of BD. if O is the mid-point
of AE, prove that ar (ΔBOE) = `1/8` ar (Δ ABC).

Answer 1

Given that
D is the midpoint of side BC of ΔABC .
E is the midpoint of BD and
O is the midpoint of AE
Since AD and AE are the medians of  ΔABC and ABD respectively

 ∴ ar (Δ ABC) = `1/2` AR (ΔABC)             ........... (1)

   ar (ΔABE) `1/2` ar (ΔABD)           ........... (2) 

  OB is a median of (ΔABE)

  ∴ ar (ΔBOE) = `1/2` ar (ΔABE)

  From 1, (2) and (3) we have

  ar (BOE) =`1/8` ar (ΔABC)