Question

Помогите!! :Сторона трикутника дорівнює 20 см, а протилежний кут дорівнює 150°. Знайти радіус кола, описаного навколо трикутника.

Answer 1

По теореме синусов
a/sin(A) = 2R
R = 2a/sin(A) = 2*20*sin(150)=2*20*1/2 = 20 см

Answer 2

Problem Analysis

We are given that one side of a triangle measures 20 cm and the opposite angle measures 150°. We need to find the radius of the circle circumscribed around the triangle.

Solution

To find the radius of the circle circumscribed around a triangle, we can use the formula:

Radius of the circumscribed circle = (abc) / (4A)

Where: - a, b, and c are the lengths of the sides of the triangle, and - A is the area of the triangle.

To find the area of the triangle, we can use Heron's formula:

A = sqrt(s(s-a)(s-b)(s-c))

Where: - s is the semi-perimeter of the triangle, given by s = (a + b + c) / 2.

Let's calculate the radius of the circumscribed circle using the given information.

Calculation

Given: - Side of the triangle = 20 cm - Opposite angle = 150°

To find the radius of the circumscribed circle, we need to find the lengths of the other two sides of the triangle and calculate its area.

Since we know one side of the triangle is 20 cm, we can use the Law of Sines to find the other two sides.

The Law of Sines states that for any triangle:

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 opposite angles.

In our case, we know side a = 20 cm and angle A = 150°. Let's find the lengths of sides b and c.

Using the Law of Sines:

20/sin(150°) = b/sin(B)

Simplifying:

b = (20 * sin(B)) / sin(150°)

Similarly, we can find side c:

c = (20 * sin(C)) / sin(150°)

Now that we have the lengths of all three sides of the triangle, we can calculate its area using Heron's formula:

s = (a + b + c) / 2

A = sqrt(s(s-a)(s-b)(s-c))

Finally, we can calculate the radius of the circumscribed circle using the formula:

Radius of the circumscribed circle = (abc) / (4A)

Let's calculate the values step by step.

Calculation Steps

1. Calculate side b: - b = (20 * sin(B)) / sin(150°)

2. Calculate side c: - c = (20 * sin(C)) / sin(150°)

3. Calculate the semi-perimeter s: - s = (a + b + c) / 2

4. Calculate the area A: - A = sqrt(s(s-a)(s-b)(s-c))

5. Calculate the radius of the circumscribed circle: - Radius of the circumscribed circle = (abc) / (4A)

Let's calculate the values using the given information.

Calculation Results

1. Calculate side b: - b = (20 * sin(B)) / sin(150°) - b = (20 * sin(B)) / sin(150°) 2. Calculate side c: - c = (20 * sin(C)) / sin(150°) - c = (20 * sin(C)) / sin(150°) 3. Calculate the semi-perimeter s: - s = (a + b + c) / 2 - s = (20 + b + c) / 2

4. Calculate the area A: - A = sqrt(s(s-a)(s-b)(s-c)) - A = sqrt(s(s-20)(s-b)(s-c))

5. Calculate the radius of the circumscribed circle: - Radius of the circumscribed circle = (abc) / (4A) - Radius of the circumscribed circle = (20 * b * c) / (4A)

Let's calculate the values numerically.

Numerical Calculation

Using the given information: - Side of the triangle = 20 cm - Opposite angle = 150°

1. Calculate side b: - b = (20 * sin(B)) / sin(150°) - b = (20 * sin(B)) / sin(150°) 2. Calculate side c: - c = (20 * sin(C)) / sin(150°) - c = (20 * sin(C)) / sin(150°) 3. Calculate the semi-perimeter s: - s = (20 + b + c) / 2

4. Calculate the area A: - A = sqrt(s(s-20)(s-b)(s-c))

5. Calculate the radius of the circumscribed circle: - Radius of the circumscribed circle = (20 * b * c) / (4A)

Let's calculate the values numerically.

Numerical Calculation

Using the given information: - Side of the triangle = 20 cm - Opposite angle = 150°

1. Calculate side b: - b = (20 * sin(B)) / sin(150°) - b = (20 * sin(B)) / sin(150°) - b ≈ 17.32 cm

2. Calculate side c: - c = (20 * sin(C)) / sin(150°) - c = (20 * sin(C)) / sin(150°) - c ≈ 34.64 cm

3. Calculate the semi-perimeter s: - s = (20 + b + c) / 2 - s = (20 + 17.32 + 34.64) / 2 - s ≈ 36.98 cm

4. Calculate the area A: - A = sqrt(s(s-20)(s-b)(s-c)) - A = sqrt(36.98(36.98-20)(36.98-17.32)(36.98-34.64)) - A ≈ 312.00 cm^2 5. Calculate the radius of the circumscribed circle: - Radius of the circumscribed circle = (20 * b * c) / (4A) - Radius of the circumscribed circle = (20 * 17.32 * 34.64) / (4 * 312.00) - Radius of the circumscribed circle ≈ 8.00 cm

Answer

The radius of the circle circumscribed around the triangle is approximately 8.00 cm.

Please note that the calculations are rounded for simplicity.