In this case, a chord parallel to the diameter is 3/4 the length of the diameter. Call the Diameter 16, the radius 8, the chord 12. From the center of the circle to the endpoint of the chord is a radius, 8. Drop a perpendicular from the endpoint of the chord to the diameter. It meets the diameter 6 units from the center. By Pythagoras, the perpendicular is 2sqr(7) units long.
Or, try this. Let the radius be 8, the perpendicular be 4, then the half-chord (by Pyth) is radical 48.