# The vertices of a triangle divide the circle described around it into three arcs, the lengths of which are referred to as 3: 4: 11

The vertices of a triangle divide the circle described around it into three arcs, the lengths of which are referred to as 3: 4: 11. Find the radius of the circle if the smaller of the sides is 14 The degree measure of the entire circumference is 360 °.
We divide it into equal conditional parts so that one arc has 3 such parts, the second arc 4 parts, and the third 11 parts (as in the condition of the problem). Then it’s clear that we need 3 + 4 + 11 such parts, a total of 18.
The degree measure of each part is 360 ° / 18 = 20 °.
Then our first arc has a degree measure of 20 ° * 3 = 60 °, the second – 20 ° * 4 = 80 °, the third – 20 ° * 11 = 220 °.
The angles ABC, BCA and CAB are inscribed in a circle, therefore, they are equal to half the degree measure of the arc on which they rely, i.e.: One angle is 30 °, the second 40 °, and the third 110 °.
By the theorem on the ratio of angles and sides of a triangle: a smaller angle lies on the opposite side. The smaller angle is 30 ° (we just calculated this), and the smaller side is 14 (according to the conditions of the problem).
By the sine theorem 14 / sin30 ° = 2R
14 / 0.5 = 2R
28 = 2R
R = 14 Remember: The process of learning a person lasts a lifetime. The value of the same knowledge for different people may be different, it is determined by their individual characteristics and needs. Therefore, knowledge is always needed at any age and position.