Question

A triangle has a 60° angle, and the two adjacent sides are 12 and "12 times the square root of 3". Find the radius of a circle with the same vertex as a center, if the arc in the triangle bisects the area of the triangle.

Answer 1

To calculate the radius of the circle, you need to use the Law of Sines. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is the same for all three sides of the triangle. Therefore, in your triangle, the ratio of the length of the side opposite the 60° angle to the sine of the 60° angle is the same as the ratio of the length of the other two sides to their respective sines.

Let's call the two adjacent sides of the triangle a and b, respectively. The Law of Sines states that:

a/sin(60°) = b/sin(30°)

The length of side a is 12, and the length of side b is 12 times the square root of 3. This means that:

12/sin(60°) = (12√3)/sin(30°)

We can rearrange this equation to calculate the radius of the circle:

r = 12 * sin(30°) / sin(60°)

This gives us a radius of 12√3.

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