The area of a right-angled triangle is 32√3 One of the acute angles is 30 °. Find the length of the leg opposite this corner.

Let’s denote by x the length of the leg of this right-angled triangle, which lies opposite the angle of 30 °, and by y – the length of the second leg.

Using the formulas of a right-angled triangle, we can express the length of the second leg through the length of the first leg:

y = x * tan 30 ° = x / √3.

By the condition of the problem, the area of ​​this right-angled triangle is 32√3.

Since the area of ​​any right-angled triangle is half the product of its legs, we can write the following equation:

(1/2) * x * x / √3 = 32√3.

Solving this equation, we get:

x ^ 2 / (2√3) = 32√3;

x ^ 2 = 32√3 * 2√3;

x ^ 2 = 32 * 2 * 3;

x ^ 2 = 64 * 3;

x = √ (64 * 3);

x = 8√3.

Answer: the length of the leg opposite the 30 ° angle is 8√3.