Question

Prove that every line segment has one and only one mid-point. Give Euclid's axiom which is used.

Answer 1

Let there be 2 midpoints C and D of line AB

When C is MP, AC =BC ---- eqn 1

When D is MP, AD = BD ---- eqn2

Subtract eqn 1 frm 2

AD - AC = BD - BC

CD= -CD (writing DC as -CD means the same)

CD + CD = 0

2CD = 0

CD = 0

This is possible only when C coincides with D

Hence there is only 1 midpoint.

So easy it is....

Answer 2

Let us consider, a line segment AB.
Assume that it has two midpoints say C and D
 
Recall that the midpoint of a line segment divides it into two equal parts
That is AC = BC and AD = DB
Since C is midpoint of AB, we have A, C and B are collinear
∴ AC + BC = AB → (1)
Similarly, we get AD + DB = AB → (2)
From (1) and (2), we get
AC + BC = AD + DB
2 AC = 2AD
∴ AC = AD
This is a contradiction unless C and D coincide.
Therefore our assumption that a line segment AB has two midpoints is incorrect.
Thus every line segment has one and only one midpoint.
I hope it is useful! !!!!
Thanku