Q. 345.0( 3 Votes )

Prove that every line segment has a unique mid-point.

Answer :

Let us consider, a line segment AB.
Assume that it has two midpoints say C and D



Midpoint of a line segment divides it into two equal parts
So, AC = BC and AD = DB
Since, C is midpoint of AB, we have A, C and B are collinear
Thus, AC + BC = AB ……………… (i)



Similarly, we get AD + DB = AB ……………(ii)


From eq(i) and (ii), 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.


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