Question

If point C is the midpoint of line segment AB. Prove that every line segment has one and only one midpoint.

Answer 1

Hint: Use the method of contradiction. Let the line segment AB have two midpoints C and D. So, AC = BC and AD = DB. The points A, B, C, and D are collinear. So, AC + BC = AB and AD + DB = AB. Equate these and then use the previous equation to get AC = AD. This is a contradiction.

Complete step-by-step solution
In this question, we are given that point C is the midpoint of line segment AB.
We need to prove that every line segment has one and only one midpoint.
We will solve this question using the method of contradiction.
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 …………………………………(1)
Since C is the midpoint of AB, we have A, C, and B are collinear
AC + BC = AB …………………………………(2)
Similarly, we get AD + DB = AB ……………………………(3)
From (2) and (3), we get the following:
AC + BC = AD + DB
Using (1), we will get the following:
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.

Note: In this question, it is very important to know what the method of contradiction is. When proving something is true using proof by contradiction, you assume the statement to be false, and as you proceed with the proof, you run into a contradiction, making the assumption of your original statement being false impossible, thus it must be true.