Question

In a line segment AB point C is called a mid-point of line segment AB. Prove that every line segment has one and only one midpoint.

Answer 1

take c(0,0) on origin and take co-ordinates of point A(-a,0) and of point B(a,0)
By applying distance formula prove that AC = CB...and hence proved, if you want complete solution I can provide it to you.

Answer 2

I have to prove it using Euclid's Axioms.

Answer 3

A line is represented by a linear equation. So when you solve this equation for midpoint you will also get a linear equation.
And you know
A linear equation has only one solution.
So you will get only one value for midpoint.

Answer 4

To prove, you need show 2 different midpoints are the same.
Let \(A=(X_A,Y_A), B=(X_B,Y_B)\)
Suppose C, C' both are midpoints of AB.
Since C is midpoint of AB, the coordinate of C is \(C =(\dfrac{X_A+X_B}{2}, \dfrac{Y_A+Y_B}{2})\)
Since C' is also midpoint of AB, the coordinate of C' is \(C'=(\dfrac{X_A+X_B}{2},\dfrac{Y_A+Y_B}{2})\)
Hence C = C'\(\implies\) there is only 1 midpoint of the segment AB