The correct option is
A True
Given :ABC is a Triangle in which the midpoints of sides BC, CA and AB are D, E and F.
To show,ar(BDEF)=1/2(ABC)
Proof:
We can see that triangle ABC and AFE are similar. Therefore,
ar(AFE)/ar(ABC)=(AF)2/(AB)2=(1/2)2=1/4
ar(AFE)=ar(ABC)/4
Similarly,
ar(CDE)=ar(ABC)/4
ar(BDF)=ar(ABC)/4 and
ar(FDE)=ar(ABC)−ar(AFE)−ar(BDF)−ar(CDE)
ar(FDE)=ar(ABC)−ar(ABC)/4−ar(ABC)/4−ar(ABC)/4
ar(FDE)=ar(ABC)/4
Thus, ar(FDE)=ar(AFE)=ar(BDF)=ar(CDE)=ar(ABC)/4
Now, according to question,
ar(ABC)=ar(FDE)+ar(AFE)+ar(BDF)+ar(CDE)
ar(ABC)=ar(BDEF)+ar(ABC)/4+ar(ABC)/4(becausear(BDEF)=ar(BDF)+ar(DEF))
ar(BDEF)=ar(ABC)−ar(ABC)/2
ar(BDEF)=ar(ABC)/2