Circle geometry problem: finding the sine of a central angle given the radius and a chord

Question

This question is for an online math class I'm taking, and I can't figure out how to do it. I'm sure I should use Power of a Point, but I don't know how to start.

Answer 1

I'm not seeing the application of the power of a point (yet), but it is possible to construct the exact arc $AB$ and to find its sine, for example by finding Cartesian coordinates of points $O,A,B,C,D$ that satisfy the conditions of the problem and then applying some analytic geometry with perpendicular lines.

Another option is to express the angle $\angle AOB$ as the difference of two other angles, where you know the sines and cosines of both of the other angles, and use the trig formula for the sine of a difference of angles. That method seems easier to me.

As an additional hint to get started, the statement that "there is only one chord starting from $A$ that is bisected by $BC$" is an important clue. From most points on the circle, there either are two chords that are bisected by $BC$ or there are no such chords.

There is only one point that can be the other end of the bisected chord. Identify that point, and you should then be able to construct the point $A$ and the arc $AB.$