Question

На колі з центром у точці О і радіусом 1 м розміщено точки А,В, С так,що АВ- діаметр кола, АС =√3м. Під яким кутом видно з центра кола хорду ВС?

Answer 1

Problem Statement

У колі з центром у точці О і радіусом 1 м розміщено точки А, В, С так, що АВ - діаметр кола, АС = √3 м. Під яким кутом видно з центра кола хорду ВС?

Solution

To find the angle at which the chord VS is visible from the center of the circle, we can use the properties of a circle and trigonometry.

Let's consider the triangle OVS, where O is the center of the circle, V is one end of the chord VS, and S is the other end of the chord. We know that AV is the diameter of the circle, so angle VOS is a right angle.

We can use the Pythagorean theorem to find the length of the chord VS. Since AV is the diameter, we have AV = 2 * OS. Using the given information, we can calculate the length of OS:

OS = √(AV^2 - AS^2) = √((2 * OS)^2 - (√3)^2) = √(4 * OS^2 - 3)

Simplifying the equation, we get:

OS^2 = (4 * OS^2 - 3) 3 * OS^2 = 3 OS^2 = 1 OS = 1

Therefore, OS = 1.

Now, we can use trigonometry to find the angle at which the chord VS is visible from the center of the circle. Let's denote this angle as θ.

In the right triangle OVS, we have:

sin(θ) = OS / OV sin(θ) = 1 / 1 sin(θ) = 1

To find the value of θ, we can take the inverse sine (arcsin) of 1:

θ = arcsin(1) θ = 90 degrees

Therefore, the chord VS is visible from the center of the circle at an angle of 90 degrees.

Answer: The chord VS is visible from the center of the circle at an angle of 90 degrees.