Converse of Mid-Point Theorem

Problem 16 : ) Prove that :-
"The line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side and half of the other side"


Solution : 


Refer to the diagram given in the link below.
http://i854.photobucket.com/albums/ab104…

Let D is the mid-point of AB of Δ ABC and
DE drawn is parallel to BC.

To Prove That : E is the mid-point of AC.

Proof :
Extend DE and draw a line // BD ( or BA) which meets the extended DE at F .
=> BCFD is a //gm. {since opp. sides are parallel }
=> AD = DB = CF .

In Δs ADE and CFE,
1) ∠ADE = ∠CFE. . . . . . (Alt ∠s)
2) ∠AED = ∠ CEF . . . . . (Vert. opp. ∠s)
3) AD = CF

=> Δ ADE ≅ Δ CFE . . . . .(By A.A.S Test)
=> AE = CE and DE = EF . . . . . . .(C.S.C.T. )

Which shows that E is the mid-point of AC .
Further, BC = FD . . . {opp. sides are equal and parallel}
and DE = (1/2) FD
=> DE = (1/2) BC. ===> (Proved)


Link to Y/A