Question

Стороны угла A касаются окружности с центром O радиуса R. Определи расстояние OA, если ∡A = 60° и R = 15 см​

Answer 1

Ответ:30

Объяснение: Из центра окружности О проведём радиус в точку касания М, тогда ОМ⊥ ОА (по свойству касательной); ⇒Δ АОМ -прямоугольный, у него ∠ОАМ =60°62=30° (по свойству касательных, проведённых из одной точки к окр); по св-ву катета, лежащего против угла в 30° ⇒ОА=2·ОМ=2·15=30

Answer 2

I can answer your question in detail. The sides of angle A touch the circle with center O and radius R. Determine the distance OA, if ∡A = 60° and R = 15 cm.

To solve this problem, we can use the property of tangents drawn from one point to a circle. According to this property, the angle between the tangents is equal to the angle subtended by the chord joining the points of contact at the center of the circle. Therefore, if we draw a chord AB through the points of contact of the tangents AC and BC, we will have ∡AOB = ∡A = 60°.

Now, we can apply the theorem of sines to the triangle AOB. The theorem states that for any triangle ABC, the ratio of the sine of an angle to the opposite side is constant. That is,

$$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$$

In our case, we have

$$\frac{\sin \angle AOB}{AB} = \frac{\sin \angle OAB}{OA} = \frac{\sin \angle OBA}{OB}$$

Since AB is a chord of the circle, we can find its length using the formula

$$AB = 2R\sin \frac{\angle AOB}{2}$$

where R is the radius of the circle. Substituting R = 15 cm and ∡AOB = 60°, we get

$$AB = 2\cdot 15\cdot \sin \frac{60}{2} = 15\sqrt{3} \text{ cm}$$

Now, we can use the first ratio in the theorem of sines to find OA. We have

$$\frac{\sin \angle AOB}{AB} = \frac{\sin \angle OAB}{OA}$$

Since ∡AOB = ∡A = 60° and ∡OAB = ∡OBA = 90° - ∡AOB/2 = 90° - 30° = 60°, we can simplify this equation as

$$\frac{\sin 60}{15\sqrt{3}} = \frac{\sin 60}{OA}$$

Cross-multiplying and solving for OA, we get

$$OA = 15\sqrt{3}\cdot \frac{\sin 60}{\sin 60} = 15\sqrt{3} \text{ cm}$$

Therefore, the distance OA is equal to 15√3 cm.

I hope this answer helps you understand the problem better.