Question

В кругу с центром в точке О проведено хорду АВ и диаметр ВС. Найдите углы треугольника АОС, если угол АОВ = 146 градусов.

Answer 1

Problem Analysis

We are given a circle with center O and a chord AB, where AB is a chord and BC is a diameter. We need to find the angles of triangle AOC, given that angle AOB is 146 degrees.

Solution

To find the angles of triangle AOC, we need to use the properties of angles in a circle.

1. Angle AOC: This angle is formed by the chord AB and the diameter BC. According to the inscribed angle theorem, the measure of an angle formed by a chord and a diameter is always 90 degrees. Therefore, angle AOC is 90 degrees.

2. Angle OAC: This angle is formed by the radius OA and the chord AB. According to the theorem, the measure of an angle formed by a radius and a chord is half the measure of the intercepted arc. In this case, angle OAC is half the measure of arc AB. Since angle AOB is given as 146 degrees, arc AB is also 146 degrees. Therefore, angle OAC is 73 degrees.

3. Angle OCA: This angle is formed by the radius OA and the diameter BC. According to the inscribed angle theorem, the measure of an angle formed by a radius and a diameter is always 90 degrees. Therefore, angle OCA is 90 degrees.

Summary

In summary, the angles of triangle AOC are as follows: - Angle AOC: 90 degrees - Angle OAC: 73 degrees - Angle OCA: 90 degrees

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