# Math Help - Simple problem, We know this but how to prove this?

1. ## Simple problem, We know this but how to prove this?

I had a question in my exam.
Code:
Prove that a line segment has only one mid point.
I wrote that its not possible to have more than one mid point as it has to be equal on both sides. i got 1 mark outta 3 marks. Can anyone gimme proper answer with proof and all?

2. You have only suggested that there must be only one. You have not proven it.

1) Draw a line segment with a midpoint.
2) Note that the two sides have identical length
3) Put another midpoint on the line segment. I doesn't matter where.
4) Note that the distance between the two midpoints measures other than zero.
5) Call the three segments so defined A, B, and C. B is between the two midpoints.
6) If both are midpoints, then mA = mB + mC AND mA + mB = mC
7) What does this say about the length of B?
8) What does that say about where the second "midpoint" really is?

3. Hmm...would you be allowed to use perpendicular bisectors?

This is how I would approach it

All points equidistant from endpoints A and B would be on the perpendicular bisector, however, only one point, the midpoint, is on the perpendicular bisector and $\overline{AB}$. Therefore exactly one midpoint exists on a line segment.