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
Answer: R = 14
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