How do you prove that a point on a perpendicular bisector is equidistant from the endpoints of the segment it intersects?

Check the picture drawn

Consider the line segment AB, 

let M be the midpoint of AB, so AM=MA

erect the perpendicular MT to AB from point M. Pick a point P on MT and join it to the points A and B

The triangles PMA and PMB are congruent from the Side Angle Side congruence postulate:

AM=MA, PM is common and m(PMA)=m(PMB)=90°, as MT is perpendicular to AB

so PA=PB