The area of a right-angled triangle is 210 ^ 2 cm square, the hypotenuse is 37 cm. Find the perimeter of this triangle.

Area of ​​a right-angled triangle S = 210 cm ^ 2, hypotenuse c = 37 cm.

The formula for the area of ​​a right-angled triangle is S = ½ * a * b.

The formula for the perimeter of a triangle is P = a + b + c = a + b + 37.

Let’s compose a system of two equations:

½ * a * b = 210;

a ^ 2 + b ^ 2 = (37) ^ 2 = 1369 (by the Pythagorean theorem).

From the second equation:

a ^ 2 + b ^ 2 = (a + b) ^ 2 – 2 * a * b = (a + b) ^ 2 – 2 * 420 = (a + b) ^ 2 – 840,

a ^ 2 + b ^ 2 = (a – b) ^ 2 + 2 * a * b = (a – b) ^ 2 + 840,

(a + b) ^ 2 = 840 + 1369 = 2209 = (47) ^ 2.

(a – b) ^ 2 = – 840 +1369 = 529 = (23) ^ 2.

Since the sum of the parties cannot be negative, therefore,

a + b = 47.

a – b = 23.

Add the last two equations,

2 * a = 47 + 23 = 70, a = 35.

Then, b = 420 / a = 420/35 = 12.

The perimeter of the triangle P = 35 + 12 + 37 = 84 (cm).