The hypotenuse of a right-angled triangle is 13 cm, and its area is 30 cm². Find the smallest leg of this triangle.

Let’s denote the legs of a right-angled triangle through x cm and y cm.The area of a right-angled triangle is half the product of the legs, hence:
x * y = 60.
By the Pythagorean theorem, x ^ 2 + y ^ 2 = 13 ^ 2.
We get the system:
{x * y = 60,
{x ^ 2 + y ^ 2 = 169.
Multiply the first equation by 2 and add to the second:
x ^ 2 + 2xy + y ^ 2 = 289,
(x + y) ^ 2 = 289.
Considering that x and y are positive numbers, we get:
x + y = 17,
x * y = 60.
Following reasoning similar to Vietta’s theorem, we obtain legs 5 cm and 12 cm.
Answer. The smaller leg is 5 cm.