Question

Точка А и окружность радиуса 6 см лежат в плоскости, расстояние от точки А до центра окружности равно 12 см. Вычислите градусную меру угла между касательными к окружности, проведёнными через точку А.

Answer 1

Если весь угол α, то для $sin( \frac{ \alpha }{2} ) = \frac{6}{12} = \frac{1}{2}$
значит $\frac{ \alpha }{2} = 30$

значит α = 2*30 =60

Answer 2

The question you asked is about finding the angle between the tangents to a circle from an external point. Here is a detailed answer:

Let O be the center of the circle and A be the external point. Let B and C be the points of contact of the tangents from A to the circle. Then, we have:

- The radius of the circle is 6 cm, so OB = OC = 6 cm. - The distance from A to O is 12 cm, so OA = 12 cm. - The triangles OAB and OAC are right triangles, because the radius is perpendicular to the tangent at the point of contact. - The angle AOB is the angle between the tangents, which we want to find.

Using the Pythagorean theorem, we can find the length of AB and AC:

AB = AC = sqrt(OA^2 - OB^2) = sqrt(12^2 - 6^2) = sqrt(108) = 6 sqrt(3) cm

Using the trigonometric ratios, we can find the angle AOB:

sin(AOB/2) = AB/OA = 6 sqrt(3) / 12 = sqrt(3) / 2

AOB/2 = 60 degrees

AOB = 120 degrees

Therefore, the angle between the tangents is 120 degrees.

You can see a diagram of the solution here: [Diagram]

This answer is based on the information from the following sources: [Source 1](https://uchi.ru/otvety/questions/tochka-a-i-okruzhnost-radiusa-6-sm-lezhat-v-ploskosti-rasstoyanie-ot-tochki-a-do-tsentra), [Source 2](https://online-otvet.ru/geometria/5cea766196f4e19a29fe2e00), [Source 3](https://www.evkova.org/zadachi-na-postroenie-po-geometrii)