The area of an isosceles triangle is 2+ roots of 3, and the angle between the sides is 30 degrees. find the radius of the circle around the triangle
The radius of a circle circumscribed around a triangle is:
R = (abc) / 2S, where a, b, c are the sides of the triangle, S is its area.
In our case, a = b, which means that the formula can be written as follows:
R = (2ac) / 2S = ac / S.
Let’s draw the height to the base. In the resulting two right-angled triangles, the hypotenuse is a, and the lower leg is 1 / 2c. Let us express the sine of the angle at the vertex:
sin A = 1 / 2c / a.
c = 2 sin A * a.
The cosine of the angle at the vertex is
cos A = h / a, where h is the height of the isosceles triangle.
Let us express h:
h = cos A * a.
The area of an isosceles triangle is:
S = ½ * c * h.
S = ½ * 2 * sin A * a * cos A * a = a² * sin A * cos A.
We substitute the found expressions for c and S into the formula for the radius of the circumscribed circle:
R = (2a * 2 * sin A * a) / (2 * a² * sin A * cos A) = 2 / cos A.
Find angle A:
Angle A = 30/2 = 15̊.
Now let’s find the radius:
R = 2 / cos 15̊ ≈ 2.07.
Answer: The radius of the circumscribed circle is approximately 2.07.