The difference between the legs of a right-angled triangle is 23 dm, and its hypotenuse is 37 dm.

The difference between the legs of a right-angled triangle is 23 dm, and its hypotenuse is 37 dm. Find the perimeter of the triangle.

Suppose that one of the legs of a given right-angled triangle is x, then the second leg, according to the condition of the problem, will be x – 23.

Let’s use the Pythagorean theorem and compose the following equation:

x² + (x – 23) ² = 37²,

x² + x² – 46 * x + 529 = 1369,

2 * x² – 46 * x – 840 = 0,

x² – 23 * x – 420 = 0.

The discriminant of this equation is:

(-23) ² – 4 * 1 * (-420) = 2209.

Since x can only be a positive number, the problem has a unique solution:

x = (23 + 47) / 2 = 35 (dm) – the length of the larger leg.

32 – 23 = 12 (dm) – the length of the smaller leg.

Therefore, the perimeter of this triangle is:

P = 37 + 35 + 12 = 84 (dm).