The area of a right-angled triangle is 128√3. One of the sharp corners is 30∘. Find the length of the leg opposite this corner.

In a right-angled triangle, we denote the hypotenuse by C.

Since the problem says that one of the angles of the triangle is 30 °, therefore, the side opposite to the corner is half this value:

The opposite leg is C / 2.

Knowing that the area of the triangle is 128√3, we compose and solve the equation according to the Pythagorean theorem:

√ (C ^ 2 + (C / 2)) ^ 2 = 128√3;

√ (3C ^ 2/4) = 128√3;

(C / 2) * √3 = 128√3;

C = (2 * 128√3) / √3;

C = (256 * √3) / √3;

C = 256.

Answer: the length of the leg opposite this 30 ° angle is 256.