Question

МНОГО БАЛЛОВ Дано: ω(O;R) AC,AB - Касательная ∠ BAC = 60° Найти : BC

Answer 1

Решение задания смотри на фотографии

Answer 2

Problem Statement

We are given a triangle ABC, where AC and AB are tangents to the circle with center O and radius r. We are also given that angle BAC is 60°. We need to find the length of BC.

Solution

To find the length of BC, we can use the properties of a tangent to a circle.

Since AC and AB are tangents to the circle, we know that the angles formed between the tangents and the radii drawn to the points of tangency are right angles. Therefore, angle BAO and angle CAO are both right angles.

Let's denote the point of tangency between AC and the circle as D, and the point of tangency between AB and the circle as E.

Since angle BAC is 60°, angle BAO is a right angle, and angle BAC + angle CAO = 90°, we can conclude that angle CAO is 30°.

Now, we can use trigonometry to find the length of BC. Let's denote the length of BC as x.

In triangle ABC, we have: - angle BAC = 60° - angle CAO = 30° - angle ABC = 90°

Using the sine rule, we can write: sin(angle BAC) / BC = sin(angle ABC) / AC

Substituting the known values, we get: sin(60°) / x = sin(90°) / r

Simplifying the equation, we have: √3 / x = 1 / r

Cross-multiplying, we get: x = r * √3

Therefore, the length of BC is r * √3.

Answer

The length of BC is r * √3.

Note: The above solution is based on the given information and the application of trigonometry.