state and prove mid-point theorem.
Answers
The line segment joining the mid-points of two sides of a triangle is parallel to the third side and equal to half the third side.
Take a triangle ABC,E and F are the mid-points of side AB and AC resp.
Construction:-Through C,draw a line II BA to meet EF produced at D.
Proof:-
In Triangle AEF and CDF
1. AF=CF(F is midpoint of AC)
2. Angle AFE= Angle CFD (Vertically opp. angles)
3. Angle EAF= Angle DCF [Alt. angles,BA II CD(by construction) and AC is a transversal]
4. So,Triangle AEF = CDF(Angle side Angle rule)
5. EF=FD AND AE = CD (c.p.c.t)
6. AE=BE(E is midpoint of AB)
7. BE=CD(from 5 and 6)
8.EBCD is a IIgm [BA II CD (by construction) and BE = CD(from 7)]
9.EF II BC AND ED=BC (Since EBCD is a IIgm)
10.EF = 1/2 ED (Since EF = FD,from 5)
11.EF = 1/2 BC (Since ED = BC,from 9)
Hence,EF II BC AND EF = 1/2 BC which proves the mid-point theorem.
(ans)..