The perimeter of the triangle is 84 cm, and its hypotenuse is 37 cm. Find the area of the triangle.

Since the perimeter of the triangle is p = r + k1 + k2 = 84 cm, where r is the hypotenuse = 37 cm, and the legs are k1 and k2.

Let’s define these legs:

k1 + k2 = p – r = 84 cm – 37 cm = 47 cm. k1 = 47 – k2.

The legs are connected by the ratio:

g² = к1² + к2² = (47 – к2) ² + (к2) ² = 2209 + (к2) ² – 2 * 47 * к2 + (к2) ² = 2209 + 2 * (к2) ² – 2 * к2 = 37² …

Let’s bring everything to the same parameters:

2 * k² – 94 * k +2209 – 1369 = 0; k² – 47 * k + 420 = 0.

к1,2 = 47/2 + – √ [(47/2) ² – 420] = 47/2 + – √ [(2209 – 420 * 4) / 4] = 47/2 + – √529 / 4 = 47 / 2 + – 23/2.

k1 = 35, k2 = 12.

Area = k1 * k2 / 2 = 12 * 35/2 = 210 cm².