Question

In Question 4, 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

Let us consider two points C and D, both supposed to be mid-points of line segment AB.

Step 1: Establishing the Equality of Segments 

Since \( C \) is the mid-point of \( AB \), it follows that:

\[ AC = BC \]

By adding \( AC \) to both sides of the equation, we get:

\[ AC + AC = BC + AC \tag{1} \]

Here, \( BC + AC \) coincides with \( AB \), which is a fundamental property of a line segment. Therefore:

\[ BC + AC = AB \tag{2} \]

By the transitive property of equality, we know that if two things are equal to the same thing, they are equal to each other. Hence, from equations (1) and (2), we get:

\[ AC + AC = AB \] \[ 2AC = AB \tag{3} \]

Step 2: Using the Mid-Point Property for Point D

Next, let us take \( D \) as another mid-point of \( AB \). By a similar argument, we can write:

\[ 2AD = AB \tag{4} \]

Step 3: Conclusion from Equations

Now, from equations (3) and (4), we can conclude that:

\[ 2AC = 2AD \]

Since things which are equal to the same thing are equal to each other, we get:

\[ AC = AD \]

Step 4: Contradiction

This result implies that \( C \) and \( D \) must represent the same point because they are both equal to the same length from point \( A \). However, this is only possible if point \( C \) and point \( D \) are the same point.

Conclusion:

Hence, our assumption that there could be two mid-points for a given line segment is incorrect. Therefore, there can be only one mid-point for any given line segment.