The length of chord RS in circle O, given the perpendicular bisector from the center and trigonometric calculations, is determined to be 6 units.
Step 1: Draw a perpendicular line from the center of the circle O to the chord RS, and label the intersection point as T. This line will bisect the chord RS and the angle POR.
Step 2: Use the fact that RS is parallel to PQ, and the alternate interior angles theorem, to conclude that the angle ROT is also 30 degrees.
Step 3: Use the Pythagorean theorem to find the length of OT, which is the radius of the circle. Since PQ is the diameter, we have OT = PQ/2 = 12/2 = 6.
Step 4: Use the trigonometric ratio of sine to find the length of RT, which is half of the length of RS. We have sin(30) = RT/OT, so RT = OT * sin(30) = 6 * 0.5 = 3.
Step 5: Double the length of RT to get the length of RS. We have RS = 2 * RT = 2 * 3 = 6.
Therefore, the length of RS is 6.