Question

Prove that every line segment has one and only one mid-point

Answer 1

Let us consider, a line segment AB.

Assume that it has two midpoints say C and D

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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.

Answer 2

Answer:

Step-by-step explanation:

AC = AB. ---------------------(i)

and AC' = AB----------------------(ii).

AC = AC' [Things which are equal to the same thing are equal to one another.]

This is possible only when C and C' coincide.

Hence every line segment has one and only one midpoint.

PROVED