Question

D ,E and F are respectively mid- points of the sides BC, CA, AB of a triangle ABC. show that BDEF is a parellelogram and area of DEF=1/4 area of ABC I

Answer 1

In triangle ABC
D, E, F are mid-points of BC, AC & AB respectively.
According to mid-point theorem, which states that any line passing through the mid-points of two sides of a triangle is parallel to the third side and half of it. 
FE║BD (BC and BD are on the same line) and FE=1/2BC 
But BD= 1/2BC
⇒BD= FE
Similarly,
ED║BF and ED=BF
This proves that BDEF is a parallelogram
⇒ar.DEF=ar.BFD, because a diagonal(FD divides a parallelogram into two triangles of equal areas)
Similarly, 
DCEF is a parallelogram
⇒ar.CED=ar.DEF=ar.BFD
Similarly,
DEAF is a parallelogram
⇒ar.AFE=ar.DEF=ar.EDC=ar.BFD
Ar.ABC=ar.DEF+ar.BFD+arDEC=ar.AFE
⇒Ar. ABC= 4*ar.DEF
∴ar.DEF= 1/4ar.ABC, hence proved