Question

окружности с центрами О О1 пересекаются в точках А и В. Каждая из этих окружностей проходит через центр другой. Найдите углы АОВ и ОАО1.

Answer 1

угол АОВ=120 градусов 
угол ОАО1=60 градусов 
по рисунку треугольники ОАО1 и ОВО1 равносторонние  

Answer 2

Problem Analysis

We are given two circles with centers O and O1 that intersect at points A and B. Each circle passes through the center of the other. We need to find the angles AOV and OAO1.

Solution

To find the angles AOV and OAO1, we can use the properties of intersecting chords and angles formed by intersecting chords.

Let's start by analyzing the given information and drawing a diagram:

- We have two circles with centers O and O1. - The circles intersect at points A and B. - Each circle passes through the center of the other.

Based on this information, we can draw the following diagram:

          O1
          |
          |
          A
         / \
        /   \
       /     \
      O-------B
       \     /
        \   /
         \ /
          O

Now, let's analyze the angles in the diagram:

- Angle AOV: This angle is formed by the radii OA and OV. Since both circles pass through the center of the other, the angle AOV is formed by the radii of the two circles. Therefore, angle AOV is equal to 90 degrees.

- Angle OAO1: This angle is formed by the radii OA and O1A. Since both circles pass through the center of the other, the angle OAO1 is formed by the radii of the two circles. Therefore, angle OAO1 is equal to 90 degrees.

In summary: - Angle AOV = 90 degrees. - Angle OAO1 = 90 degrees.

Please let me know if anything is unclear or if you need further assistance!