1.

State and prove mid-point theorem.​

Answer»

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)..


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