Final answer:
The radius of circle O can be determined by using the fact that angle C is inscribed in circle O and AB is a diameter of circle O. Since angle C is inscribed in circle O and AB is a diameter, angle C is the angle formed by intersection of an arc with a chord of the circle, which is half the measure of the intercepted arc. The circumference of a circle is given by the formula C = 2πr, where r is the radius of the circle. Therefore, the radius of circle O is 180/π.
Step-by-step explanation:
The radius of circle O can be determined by using the fact that angle C is inscribed in circle O and AB is a diameter of circle O.
When a diameter of a circle is formed, it cuts the circle into two equal parts, making it a right angle. Therefore, angle C is a right angle.
Since angle C is inscribed in circle O and AB is a diameter, angle C is the angle formed by intersection of an arc with a chord of the circle, which is half the measure of the intercepted arc. So, angle C is half the measure of the arc AB.
Since angle C is a right angle, we can express the measure of angle C as 90 degrees. Therefore, the measure of arc AB is 2 times angle C, which is 2 times 90 degrees or 180 degrees.
The circumference of a circle is given by the formula C = 2πr, where r is the radius of the circle. The measure of an arc is given by the formula L = 2πr * (m/360), where L is the length of the arc, r is the radius of the circle, and m is the measure of the arc in degrees.
Since the measure of arc AB is 180 degrees, we can substitute the given information into the formula to solve for the radius:
L = 2πr * (m/360)
180 = 2πr * (180/360)
180 = πr
r = 180/π
Therefore, the radius of circle O is 180/π.