In a right-angled triangle, the height drawn from the top of the right-angled angle divides the hypotenuse

In a right-angled triangle, the height drawn from the top of the right-angled angle divides the hypotenuse into two parts 9 cm and 36 cm find the perimeter of the triangle.

Let one of the legs be equal to x, the other equal to y. Because the sum of the squares of the legs is equal to the square of the hypotenuse, we can write:
x ^ 2 + y ^ 2 = 45 ^ 2 = 2025.
The height h, drawn from a right angle, divides this triangle into two right-angled triangles, in which the hypotenuses are the legs of this triangle, one of the legs is common is the height h drawn to the hypotenuse of this triangle, and the second legs are the projections of the legs of this triangle onto the hypotenuse, equal to 9 cm and 36 cm.
For each resulting triangle, we can write:
h ^ 2 = x ^ 2-9 ^ 2 = x ^ 2-81 and h ^ 2 = y ^ 2-36 ^ 2 = y ^ 2-1296.
Since h for these triangles is a common side, then x ^ 2-81 = y ^ 2-1296, x ^ 2-y ^ 2 = -1215.
Thus, we have a system of equations:
1) x ^ 2 + y ^ 2 = 45 ^ 2 = 2025;
2) x ^ 2-y ^ 2 = -1215.
Solving the system by the method of algebraic addition, we get:
x ^ 2 + y ^ 2 + x ^ 2-y ^ 2 = 2025-1215;
2 * x ^ 2 = 810;
x ^ 2 = 405;
x = √405 = 9√5.
Substituting the found value of x into any of the equations of the system, we obtain:
y ^ 2 = 1620;
y = 18√5.
The desired perimeter of the triangle is 9√5 + 18√5 + 9 + 36 = 27√5 + 45≈105.37 cm.