Question

if a point c is called a mid-point of line segment AB. prove that every line segment has one and only one mid-point

Answer 1

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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.