Question

In circle O, AC and BD are diameters.

Circle O is shown. Line segments B D and A C are diameters. A radius is drawn to cut angle C O C into 2 equal angle measures of x. Angles A O C and B O C also have angle measure x.

What is mArc A B?

72°
108°
120°
144°

Answer 1

Let AB = CD = s and BC = DA = t (opposite arcs are equal because AC and BD are diameters). Since AC is a diameter, s + t = 180°.

Angle COD subtends arc CD, so COD = s. The radius splits COD into two equal angles x, so s = 2x. Also BOC subtends arc BC, so BOC = t, and we are told BOC = x, hence t = x.

Thus 2x + x = 180° → 3x = 180° → x = 60°. Therefore arc AB = s = 2x = 120°.

Answer: 120°.