let the chord be segment AB. let the circle center be point O.
let M be the intersection point of the bisecting radius and chord AB.
M is the midpoint of AB ... why?
now, prove that triangle OMA is congruent to triangle OMB, then use corresponding parts of congruent triangles to show that angle OMA is congruent to angle OMB.
from this point, it should be easy to show that OM is perpendicular to AB.