Question

state and prove midpoint theorem​

Answer 1

The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and is equal to half of it.

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.<AFE=<CFD (Vertically opp. angles)

3.<EAF=<DCF [Alt. angles,BA II CD(by construction) and AC is a transversal]

4.So,Triangle AEF = CDF(ASA)

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.