Question

Точки А и В делят окружность с центром О на дуги АМВ и АСВ так, что дуга АСВ на 60°С меньше дуги АМВ, АМ-диаметр окружности. Найдите величины углов АМВ,АВМ,АСВ?

Answer 1

Так как дуга АСВ на 60 градусов меньше дуги АМВ, то

2АСВ+60=360

АСВ=150 градусов

АМВ=210 градусов.

Вписаный угол АМВ опирается на дугу АСВ и равен 1/2АСВ=75 градусов

Вписаный угол АВМ опирается на диаметр и равен 90 градусов

Вписаный угол АСВ опирается на дугу АМВ и равен 1/2АМВ=105 градусов

Answer 2

Problem Analysis

We are given a circle with center O and points A and B dividing the circle into arcs AMB and ACB. It is mentioned that arc ACB is 60 degrees smaller than arc AMB, and AM is the diameter of the circle. We need to find the measures of angles AMB, ABM, and ACB.

Solution

Let's analyze the given information and solve the problem step by step.

1. We are given that arc ACB is 60 degrees smaller than arc AMB. This means that the measure of arc ACB is 360 degrees - 60 degrees = 300 degrees.

2. Since AM is the diameter of the circle, angle AMB is a right angle (90 degrees).

3. Angle ABM can be found by subtracting the measure of arc ACB from the measure of angle AMB. Therefore, angle ABM = angle AMB - arc ACB = 90 degrees - 300 degrees = -210 degrees.

4. However, angles are usually measured between 0 and 360 degrees. To convert angle ABM to a positive value within this range, we can add 360 degrees to -210 degrees. Therefore, angle ABM = -210 degrees + 360 degrees = 150 degrees.

5. Finally, angle ACB can be found by subtracting angle ABM from angle AMB. Therefore, angle ACB = angle AMB - angle ABM = 90 degrees - 150 degrees = -60 degrees.

6. Similar to step 4, we need to convert angle ACB to a positive value within the range of 0 to 360 degrees. Adding 360 degrees to -60 degrees gives us angle ACB = -60 degrees + 360 degrees = 300 degrees.

Summary

The measures of the angles in the given problem are as follows: - Angle AMB = 90 degrees - Angle ABM = 150 degrees - Angle ACB = 300 degrees

Please let me know if I can help you with anything else.