The perimeter of the rectangle is 34 and its diagonal is 13, find the sides of the rectangle

The perimeter of the rectangle is 34 and its diagonal is 13, find the sides of the rectangle, the problem is solved by the system.

Let the first side be x cm, and the second side y cm.
Let’s compose a system of equations:
2x + 2y = 34;
x ^ 2 + y ^ 2 = 132;
Let’s simplify the first equation:
x + y = 17.
x = 17 – y.
Substitute the value of x in the equation x ^ 2 + y ^ 2 = 169.
(17 – y) ^ 2 + y ^ 2 = 169;
172 – 2 * 17 * y + y ^ 2 + y ^ 2 = 169;
289 – 34y + 2y ^ 2 = 169;
y ^ 2 – 17y + 60 = 0;
We have a = 1, b = – 17, c = 60.
D = b ^ 2 – 4ac = (-17) ^ 2 – 4 * 60 = 289 – 240 = 49.
x1 = (17 + 7): 2 = 12 cm.
x2 = (17 – 7): 2 = 5 cm.
Hence y1 = 17 – 12 = 5 cm.
y2 = 17 – 5 = 12 cm.
Answer: the sides can be 12 and 5 cm or 5 and 12 cm.