D,E,F are the midpoints of the sides Ab,bC,ca respectively of triangle abc.prove DEF is a parallelogram ?

Question

D,E,F are the midpoints of the sides Ab,bC,ca respectively of triangle abc.prove DEF is a parallelogram ?

Answer 1

i dont get u a parallelogram has 4 points u have mentioned only 3 pt DEF

Answer 2

I suppose ur Question is Prove BDFE a ||gm

By midpoint theorem DF=BE=BC/2 and DF||BE

Thus BDFE is a ||gm

If u wanna know bout midpt theorem, go on....

It states that

In a triangle Line joining midpts of 2 sides is || to third side and equal to half of it.

Proof:

Consider the situation you have given.

Extend DF to X such that DF = FX . Now join CX

AF=FC

DF=FX

Angle AFD = Angle CFX (Vertically opposite)

Thus Triangle AFD is congruent to Triangle CFX (by SAS)

CX = AD = BD                                 (CPCTC)

Angle ADF = Angle FXC                   (    l l    )

But these are interior opposite Angles

So AB||CX

BD||CX and BD =CX

SO BDXC is a ||gm

=> DF||BC  and DX = BC

DX/2 = BC/2

=> DF = BC/2

DF||BC and DF=BC/2   (BDFE is a ||gm)