Question

Вершины треугольника делят описанную около него окружность на три дуги, длины которых относятся как 1:2:3. Найдите радиус окружности, если меньшая из сторон равна 17. Помогите решить, пожалуйста, если можно с рисунком.

Answer 1

Вписанный треугольник АВС в окружность с центром О.
Градусная мера всей окружности 360°.
Найдем градусные меры трех дуг, для этого обозначим одну часть через х, получится уравнение:
х+2х+3х=360
х=360/6=60°
Получается градусная мера дуги АВ=60°, дуги ВС=120°, дуги АС= 180°.
Углы АВС, ВСА и САВ являются вписанными углами (вершины их лежат на окружности, а обе стороны пересекают эту окружность). Градусная мера вписанного угла равна половине дуги, на которую он опирается.
<АВС =180/2=90°, <ВСА =60/2=30° и <САВ =120/2=60°.
Исходя из того, что <АВС =90°, делаем вывод, что ΔАВС - прямоугольный и гипотенуза АС является диаметром окружности (вписанный угол, опирающийся на диаметр - прямой).
Напротив меньшей стороны лежит меньший угол, значит катет АВ=17.
Катет, лежащий против угла 30°, равен половине гипотенузы, следовательно радиус окружности ОА=ОВ=ОС=АВ=17
Ответ: 17

Answer 2

Problem Analysis

We are given a triangle inscribed in a circle, and the lengths of the arcs formed by the vertices of the triangle on the circumference of the circle. We need to find the radius of the circle.

Let's denote the lengths of the arcs as a, b, and c, with a:b:c = 1:2:3. We are also given that the smaller side of the triangle is 17 units long.

To solve this problem, we can use the fact that the ratio of the lengths of the arcs is equal to the ratio of the angles subtended by those arcs at the center of the circle. We can then use the Law of Sines to find the angles of the triangle, and finally, use the Law of Cosines to find the radius of the circle.

Solution

1. Let's denote the radius of the circle as R. 2. We know that the ratio of the lengths of the arcs is equal to the ratio of the angles subtended by those arcs at the center of the circle. Therefore, we can write the following equations: - a/R = angle A - b/R = angle B - c/R = angle C 3. Since a:b:c = 1:2:3, we can write: - a = x - b = 2x - c = 3x where x is a constant. 4. We can use the Law of Sines to find the angles of the triangle. The Law of Sines states that for any triangle with sides a, b, and c, and opposite angles A, B, and C, the following relationship holds: - sin(A)/a = sin(B)/b = sin(C)/c 5. Applying the Law of Sines to our triangle, we get: - sin(A)/x = sin(B)/(2x) = sin(C)/(3x) 6. We also know that the sum of the angles of a triangle is 180 degrees. Therefore, we have: - A + B + C = 180 7. Substituting the values of the angles from step 2 into the equation from step 6, we get: - a/R + b/R + c/R = 180 - (x/R) + (2x/R) + (3x/R) = 180 - 6x/R = 180 - x/R = 30 - x = 30R 8. Now, we can substitute the values of a, b, and c from step 3 into the equation from step 5, and simplify: - sin(A)/(30R) = sin(B)/(60R) = sin(C)/(90R) - sin(A)/30 = sin(B)/60 = sin(C)/90 - sin(A) = 30sin(B) = 2sin(C) 9. We can solve this system of equations to find the values of sin(A), sin(B), and sin(C). Let's denote sin(A) as p, sin(B) as q, and sin(C) as r. - p = 30q = 2r 10. Since sin^2(A) + sin^2(B) + sin^2(C) = 1, we have: - p^2 + q^2 + r^2 = 1 - (30q)^2 + q^2 + (2r)^2 = 1 - 900q^2 + q^2 + 4r^2 = 1 - 901q^2 + 4r^2 = 1 11. We can solve this equation to find the values of q and r. Let's denote q^2 as s. - 901s + 4r^2 = 1 12. We also know that p = 30q and r = 2p/30. Substituting these values into the equation from step 11, we get: - 901s + 4(2p/30)^2 = 1 - 901s + 4p^2/225 = 1 - 901s + 4p^2 = 225 - 901s = 225 - 4p^2 - s = (225 - 4p^2)/901 13. We can substitute the value of s into the equation p = 30q to find the value of p. Let's denote p^2 as t. - t = 30(225 - 4p^2)/901 - 901t = 30(225 - 4p^2) - 901t = 6750 - 120p^2 - 120p^2 = 6750 - 901t - p^2 = (6750 - 901t)/120 14. We can solve this equation to find the value of p. Let's denote (6750 - 901t)/120 as u. - u = p^2 - p = sqrt(u) 15. Now that we have the value of p, we can find the values of q and r using the equations from step 9. - q = p/30 - r = 2p/30 16. Finally, we can use the Law of Cosines to find the radius of the circle. The Law of Cosines states that for any triangle with sides a, b, and c, and opposite angles A, B, and C, the following relationship holds: - c^2 = a^2 + b^2 - 2ab*cos(C) 17. Applying the Law of Cosines to our triangle, we get: - R^2 = (30R)^2 + (60R)^2 - 2(30R)(60R)*cos(90R) - R^2 = 900R^2 + 3600R^2 - 3600R^2*cos(90R) - R^2 = 900R^2 + 3600R^2 - 3600R^2*0 - R^2 = 900R^2 + 3600R^2 - R^2 = 4500R^2 - 1 = 4500 - R^2 = 1/4500 - R = sqrt(1/4500) - R = 1/(30*sqrt(5))

Answer

The radius of the circle is 1/(30*sqrt(5)).

Please note that the above solution is based on the given information and assumptions made during the problem-solving process.