Does a diameter that is perpendicular to a chord bisect the chord?

Yes, any diameter which is perpendicular to a chord bisects said chord.

This can be proved most easily with a picture, but is proved using a congruent triangle proof. Both triangles include the points at the center of the circle and the intersection of the diameter and chord. The other points should be the endpoints of the chord. They are congruent by hypotenuse leg; it was given that they are right triangle by the "perpendicular", the "leg" is the segment between the center of the circle and the intersection, and it is equal in both triangles because it is the same segment in both triangles. The hypotenuses are equal because both are radii of the circle.

Because the triangles are congruent, their sides must be so the two halves of the chord are congruent, and therefore the chord is bisected by the diameter.