Points A, B, C are taken on the circle centered at point O so that arc AB refers to arc BC

Points A, B, C are taken on the circle centered at point O so that arc AB refers to arc BC as 11:12. The COA is 130 °. Find the angles BCA, BAC.

Since the central angle AOC = 130, the degree measure of the arc ABC is (360 – 130) = 230.

Then the sum of arcs (AB + BC) = 230. (1)

The ratio of arcs AB / BC = 11/12. (2)

Let’s solve the system of two equations 1 and 2.

AB = 230 – BC.

(230 – BC) / BC = 11/12.

11 * BC = 1560 – 12 * BC.

23 * BC = 2760.

BC = 2760/23 = 120.

Then the arc AB = 230 – 120 = 110.

The inscribed angle BAC rests on the arc BC, then the angle BAC = 120/2 = 60.

The inscribed BCA angle rests on the AB arc, then the BCA angle = 110/2 = 55.

Answer: The BAC angle is 60, the BCA angle is 55.