The area of an isosceles triangle with an angle at the base of 30 degrees is 64√3 cm2. find the sides of the triangle.

First, let’s find the angle between the sides of this isosceles triangle.

According to the condition of the problem, the angle at the base of this triangle is 30 °.

Since in an isosceles triangle the angles at the base are equal, and the sum of all the angles of any triangle is 180 °, the angle between the sides of this isosceles triangle is 180 – 30 – 30 = 150 – 30 = 120 °.

Let x denote the length of the lateral side of this triangle.

According to the condition of the problem, the area of ​​this triangle is 64√3 cm ^ 2, therefore, we can draw up the following equation:

x * x * sin (120 °) / 2 = 64√3,

solving which, we get:

x ^ 2 * √3 / 4 = 64√3;

x ^ 2 = 64√3 / (√3 / 4);

x ^ 2 = 64 * 4;

x = √ (64 * 4) = 8 * 2 = 16.

Using the cosine theorem, we find the length of the base of a given triangle:

√ (16 ^ 2 + 16 ^ 2 – 2 * 16 ^ 2 * cos (120 °)) = √ (256 + 256 + 512 * 0.5) = √ (256 + 256 + 256) = √ (256 * 3) = 16√3.

Answer: 16, 16 and 16√3.