Question

If D,E,F are the mid-points of the sides BC, Ca and AB respectively of a triangle ABC, prove by vector method that Area of `Delta DEF` = 1/4 Area of `Delta ABC`

Answer 1

In `Delta ABC, D and F` are the midpoints of side sAB and CA respectively.
`:. DF||BC` [ by midpoind theorem]
`rArr DF||BE`.
Similarity, `EF||BD`.
`:. BEFD` is parallelogram

`rArr angle B= angle EFD, EF= BD=(1)/(2) AB`
and `DF=BE=(1)/(2) BC`.
Also, ECFD is parallelogram
`rArr angle EDF=angle B`
and `angle EDF= angle C`
`:. Delta DEF~ Delta CAB` [ by AA- similarity]
And so, `(ar (Delta DEF))/(ar (Delta ABC))=(ar (Delta DEF))/(ar (Delta CAB))= (DF^(2))/(BC^(2))=(((1)/(2)BC)^(2))/(BC^(2))=(1)/(4)`