Question

Найдите радиус окружности, описанной около треугольника ABC, если известно, что сторона АВ=3, угол АСВ=30 градусов

Answer 1

По следствию из теоремы синусов а/(sina)=2R, R - радиус описанной окружности, получаем: 2R=3/(sin(30))=6, тогда искомый радиус равен 3

Answer 2

Finding the Radius of the Circumscribed Circle

To find the radius of the circle circumscribed around triangle ABC, we can use the given information that side AB = 3 and angle ACB = 30 degrees.

Step 1: Calculate the Length of Side AC

From the given information, we can use the law of sines to find the length of side AC. The law of sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant.

Let's denote the length of side AC as c and the length of side BC as b. Then, the law of sines can be expressed as:

a/sin(A) = b/sin(B) = c/sin(C)

Where a, b, and c are the lengths of the sides of the triangle, and A, B, and C are the measures of the angles opposite to the respective sides.

Using the given angle ACB = 30 degrees and side AB = 3, we can find the length of side AC using the law of sines.

Step 2: Calculate the Radius of the Circumscribed Circle

Once we have the length of side AC, we can use this information to calculate the radius of the circumscribed circle using the formula:

R = (a*b*c)/(4*Area)

Where R is the radius of the circumscribed circle, a, b, and c are the lengths of the sides of the triangle, and Area is the area of the triangle.

Let's proceed with the calculations based on the given information.

Calculation

Using the law of sines, we can find the length of side AC:

a/sin(A) = b/sin(B) = c/sin(C)

From the given information, side AB = 3 and angle ACB = 30 degrees.

c/sin(30) = 3/sin(B)

From the law of sines, we can solve for the length of side AC, denoted as c.

c = (3*sin(30))/sin(B)

Now, we can use the calculated length of side AC to find the radius of the circumscribed circle using the formula:

R = (a*b*c)/(4*Area)

Where a = 3, b = c, and Area can be calculated using Heron's formula or other methods.

Let's proceed with the calculations to find the radius of the circumscribed circle.