Question

prove that in a triangle the line segment joining the midpoints of any two sides is parallel to third side and is half of it.

Answer 1

Hii dear here is your answer hope it helps you!!

Answer 2

Given : 
A triangle ABC,E and F are mid points of side AB and AC respectively.
To prove :
line joining the mid points is parellel to the third side => EF || BC
and,
line joinining the mid points = half of third line  
=> EF = 1/2 BC
Construction :
Through C, draw a line parellel to BA to meet EF produced at D.
 statements                                                                 reasons
In triangle AEF and triangle CDF
1. AF = CF                                                1. F is mid point of AC(given)
2. angle AFE =  angle CFD                      2. Vertically opposite angles
3.angle EAF = angle DCF                        3.Alternate angles, BA || CD(by                                                                            construction) and AC is a                                                                                 transversa.
4.triangle AEF congruent to                    4. ASA rule of congruency
triangle CDF                                             
5.EF = FD and AE = CD                         5. c.p.c.t.
6.AE = BE                                               6. E is the mid point of AB (given)
7.BE = CD                                              7. From 5 and 6
8.EBCD is a parallelogram                     8.BA || CD (construction)
                                                                   and BE = CD (given)
9.EF || BC and ED = BC                         9.Since EBCD is a parallelogram
10. EF = 1/2 ED                                     10.Since EF = FD,from 5
11.EF = 1/2 BC                                      11.Since ED = BC , from 9

Hence, EF || BC and EF = 1/2 BC.