The perimeter of a right-angled triangle is 24cm, and its hypothesis is 10cm. Find the legs of this triangle.

Let one leg be x cm, the second y cm.
Taking into account the condition of the problem, we compose a system of equations.
x ^ 2 + y ^ 2 = 100;
x + y + 10 = 24;
We express in the second equation x through y.
x ^ 2 + y ^ 2 = 100;
x = 14-y;
Substitute the expression for x in the first equation.
(14-y) ^ 2 + y ^ 2 = 100;
196-28y + y ^ 2 + y ^ 2 = 100;
2y ^ 2-28y + 96 = 0;
y ^ 2-14y + 48 = 0;
y1 = 6; y2 = 8;
x1 = 14-6 = 8;
x2 = 14-8 = 6;
System solution: (8; 6); (6; 8).
Answer: the legs of the triangle are 6 cm and 8 cm or 8 cm and 6 cm.