The area of a right-angled triangle is 90 cm2. the sum of the areas of the squares built

The area of a right-angled triangle is 90 cm2. the sum of the areas of the squares built on its legs is 369 cm2. what are the legs of this triangle?

Let’s denote the legs (in cm) of a right-angled triangle through x and y. Then the area (S) of this right-angled triangle can be determined by the formula S = ½ * x * y.
According to the conditions of the assignment, S = 90 cm2. Therefore, ½ * (x cm) * (y cm) = 90 cm2, whence x * y = 180. This is the first equation.
According to another condition of the task, x ^ 2 + y ^ 2 = 369. This is the second equation.
Multiply both sides of the first equation by 2 and add the corresponding sides of both equations. Then, we get 2 * x * y + x ^ 2 + y ^ 2 = 2 * 180 + 369 or x2 + 2 * x * y + y2 = 729.
Applying the abbreviated multiplication formula (a + b) ^ 2 = a ^ 2 + 2 * a * b + b ^ 2 (the square of the sum), we get: (x + y) ^ 2 = 729, whence x + y = 27. This is third equation.
The first and third equations, according to Vieta’s theorem, allow us to assert that x and y are solutions to the quadratic equation z ^ 2 – 27 * z + 180 = 0. Solving this quadratic equation, we find its roots: z1 = 12 and z2 = 15.
Thus, the legs of this right-angled triangle are 12 cm and 15 cm.
Answer: 12 cm and 15 cm.