Find the area of a right-angled triangle whose perimeter is 84 and the hypotenuse is 27.

Let’s denote by x and at the leg of this right-angled triangle.

According to the condition of the problem, the hypotenuse of this right-angled triangle is 27, and the perimeter of this triangle is 84, therefore, the following relation holds:

x + y + 27 = 84,

whence follows:

x + y = 84 – 27 = 57.

Let’s square both sides of the resulting ratio:

(x + y) ^ 2 = 57 ^ 2;

x ^ 2 + 2xy + y ^ 2 = 3249;

x ^ 2 + y ^ 2 + 2xy = 3249.

Applying the Pythagorean theorem, we get:

x ^ 2 + y ^ 2 = 27 ^ 2.

Substituting this relation into the relation x ^ 2 + y ^ 2 + 2xy = 3249, we obtain:

27 ^ 2 + 2xy = 3249;

729 + 2xy = 3249;

2xy = 3249 – 729;

2xy = 2520;

2xy / 4 = 2520/4;

xy / 2 = 630.

Since the area of ​​any right-angled triangle is half the product of its legs, the area of ​​this right-angled triangle is 630.

Answer: 630.