5. D, E and F are respectively the mid-points of the sides BC, CA and AB of a .d ABC. Show that (i) BDEF is a parallel,ogram (ii) ar(])EFJ = anABCJ (iii) ar(BDEFJ = !_ar(ABCJ.

(i) Proof: Since E and F are respectively the mid-points of AC and AB of !!ABC. . 1 E • EF ll BC and EF = 2 Bc [Mid-point theorem] B 0 c Also, BD = iBC [•: D is mid-point of BC] • EF II BD and EF = BD. => BDEF is a parallelogram. [In a quadrilateral, if a pair of Op(>Osite sides is equal and parallel, then it is a parallelogram.] (ii) DF is diagonal. I :. ar ar(DEF) = 4ar(ABC). (iii) ar(BDEF) = ar(BDF) + ar(DEF). = 2.cu(DEF) [From (i)J = 2 x °14 ar(ABC) = 21 (ABC).