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

Answer: the measure of arc A B is 180°.

Explanation:

mArc A B = mArc B O + mArc O A Because line segments B D and A C are diameters, they are also chords of the circle and therefore bisect each other. This means that angle A O C and angle B O C both measure x.

A radius drawn to cut angle C O C into 2 equal angle measures of x, it means that angle C O C measure 2x. So, arcs A O and B O are also equal.

Since angle A O C and angle B O C both measure x, and arcs A O and B O are equal, it means that mArc A B = mArc B O + mArc O A = x + x = 2x

As angle C O C measure 2x, and x angle measures are equal to each other. It means that x = (2x)/2 = x = angle C O C/2 = (180°)/2 = 90°

So, mArc A B = 2x = 2(90°) = 180°

Therefore, the measure of arc A B is 180°.