Short Answer 
The process involves visualizing a triangle formed by the circle’s center and two arc intersection points, leading to a kite configuration. By applying the properties of a 30-60-90 triangle, the radius of the circle is calculated to be 6 times the square root of 3 units.
Step 1: Understand the Triangle Configuration
Firstly, visualize the triangle formed by the center of the circle and the two points where the arc intersects the triangle’s sides. The triangle consists of vertices A, B, and C, with angle CAB measuring 60¬¨‚àû. There’s also a midpoint D on side AB and intersection points E and F on sides AC and BC, respectively. The shape formed by these points is a kite BCEF with BC serving as its axis of symmetry.
Step 2: Analyze the Kite and the Midpoint Line
Next, note that the line connecting the center of the circle to the midpoint D bisects the base AB, creating equal areas on both sides of the triangle. This line segment DE, which connects the center and D, also defines the radius of the circle. Given that angle CED is 30°, we can use properties of a 30-60-90 triangle to analyze the relationship between the radius and the lengths of the sides involved.
Step 3: Calculate the Radius of the Circle
To find the radius, we utilize the side ratios in a 30-60-90 triangle, where the lengths are in the ratio of 1:‚ao3:2. Since DE is opposite the 30¬∞ angle, we can express its length as half of BC. With BC given as (12 times sqrt{3}), we calculate DE as follows:
- DE = ( frac{BC}{2} = frac{12 times sqrt{3}}{2} = 6 times sqrt{3} )
Thus, the radius of the circle is determined to be 6 times the square root of 3 units.