Question

In a circle with an 8-inch radius, a central angle has a measure of 60°. How long is the segment joining the endpoints of the arc cut off by the angle?

8
8√2
8√3

Answer 1

The other side is 8 inches.
I wish I could add a picture to this to help explain but here's my best shot.
If you draw the 60 degree angle with one side as part of a diameter it helps. Make the diameter extend across the circle, and the other side of the angle as well. Connect the lines so that both sides have a segment cutting off the arc that is congruent to the central angle. Then you have a shape almost like a bow inscribed in a circle. The 2 arcs that these angles cut off are both 60 degrees, and because we know that half of a circle is 180 degrees and 180-60 is 120, this means that the other 2 arcs left are both 120 degrees. Now back to the two triangles we created at the start (the two parts of the bow). Besides the central angle, pick one of the others (it doesn't matter which one), and look at the arc the total angle cuts off. It is the arc we already marked as 120 degrees. Because any inscribed angle in a circle is equal to half of the arc it intercepts, or "cuts off" as you put it, we know this angle also equals 60 degrees. Because a triangle contains a total of 180 degrees and we have already used 120 degrees of it (60+60), the other angle must also be 60 degrees. Because it is equiangular, this means the triangle must have equal sides. Because the original central angle is made of two radii, both equaling 8 inches, this means the other side of the triangle created must be 8 inches also.