# The vertices of a triangle divide the circle described near it into three arcs, the lengths of which are 3: 7: 8

The vertices of a triangle divide the circle described near it into three arcs, the lengths of which are 3: 7: 8. Find the radius of the circle if the smaller of the sides is 20

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 has 7 parts, and the third 8 parts (as in the condition of the problem). Then it’s clear that we need 3 + 7 + 8 of these 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 ° * 7 = 140 °, the third – 20 ° * 8 = 160 °.
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 70 °, and the third 80 °.
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 ° (this is what we just calculated), and the smaller side is 20 (by the condition of the problem).
By the sine theorem 20 / sin30 ° = 2R
20 / 0.5 = 2R
40 = 2R
R = 20