In a right-angled triangle, the area is 72√3, one of the acute angles is 30 degrees. Find the length of the hypotenuse.

Let the hypotenuse AC be equal to 2 x, and the leg BC, which lies opposite an angle of 30 °, is x.
By the Pythagorean theorem:
AC ^ 2 = AB ^ 2 + BC ^ 2
(2 x) ^ 2 = AB ^ 2 + x ^ 2
4x ^ 2 = AB ^ 2 + x ^ 2
AB ^ 2 = 4x ^ 2-x ^ 2
AB ^ 2 = 3x ^ 2
AB = x * √3
S = 1/2 * AB * BC
72 √3 = 1/2 * x * √ 3 * x
72 √3 = √ 3 * x ^ 2/2
x ^ 2 = 72 √ 3 * 2 / √ 3
x ^ 2 = 144
x = 12
BC = 12
Then AC is equal to: AC = 2 * BC
AC = 2 * 12 = 24
Answer: AC = 24.