The area of a right-angled triangle is 90 cm ^ 2, and the hypotenuse is 3√41 cm. Find the legs.

Let us denote (in centimeters) through a, b – legs and through c – the hypotenuse of this right-angled triangle.
As you know, the area of ​​a right-angled triangle is equal to half the product of the legs of the triangle, that is, (½) * a * b = 90 or a * b = 180.
According to the Pythagorean theorem, a² + b² = c². By the condition of setting the hypotenuse of the triangle is equal to 3√ (41) cm. So, a² + b² = (3√ (41)) ² or a² + b² = 369.
Thus, we got the following system of equations of the second degree: a * b = 180 and a² + b² = 369. Let’s solve this system. Multiply the first equation by 2 and find the algebraic sum of the resulting and the second equation: a² + 2 * a * b + b² = 369 + 2 * 180 or, using the abbreviated multiplication formula (a + b) 2 = a2 + 2 * a * b + b2 (the square of the sum), (a + b) 2 = 27². Taking the arithmetic square root on both sides of the last equality, we get a + b = 27.
The equalities a + b = 27 and a * b = 180, according to Vieta’s theorem, allow us to assert that a and b are the roots of the quadratic equation x² – 27 * x + 180 = 0. Let’s solve it. For this purpose, we calculate the discriminant D = (–27) ² – 4 * 1 * 180 = 729 – 720 = 9> 0. This means that the quadratic equation has two real roots: х1 = (27 – √ (9)) / 2 = 12 and x2 = (27 + √ (9)) / 2 = 15.
Thus, the legs of this triangle are 12 cm and 15 cm.
Answer: 12 cm and 15 cm.