Arc ACB = 124 degrees, arc ABD = 304 degrees, DF = 16 cm, BF = 30 cm. Circle radius =?

We connect points A, B, D and F with the exact O, the center of the circle.

Then the degree measure of the central angle AOB is equal to the degree measure of the arc ACB. Angle AOB = 124.

Similarly, the angle (larger) AOD = 304.

Determine the value of the central angle BOD.

Angle BOD = AOD – AOB = 304 – 124 = 180.

Then the segment BD is the diameter of the circle.

The inscribed angle BFD is equal to half the degree measure of the central angle of the BOD.

Then the angle BFD = 180/2 = 90, and the triangle BFD is right-angled.

By the Pythagorean theorem, BD ^ 2 = BF ^ 2 + DF ^ 2 = 900 + 256 = 1156.

ВD = 34 cm.

Then ОВ = ОD = R = ВD / 2 = 34/2 = 17 cm.

Answer: The radius of the circle is 17 cm.