The perimeter of a right-angled triangle is 84 cm, its hypotenuse is 37 cm. Find the area of this triangle.

It is known that the perimeter of a right-angled triangle is P = 84 cm, and its hypotenuse is C = 37 cm.

It is necessary to calculate the area of ​​a right-angled triangle S.

Let’s designate legs A and B.

Then we can write the following formulas for the perimeter and area, as well as the hypotenuse.

P = 2 * (A + B), from which follows A + B = P / 2, A + B = 42.

S = ½ * A * B.

C ^ 2 = A ^ 2 + B ^ 2, whence it follows that A ^ 2 + B ^ 2 = 37 ^ 2, A ^ 2 + B ^ 2 = 1369.

Let’s solve the system of equations.

1) A + B = 42.

2) A ^ 2 + B ^ 2 = 1369.

Let’s square the first equation and subtract the second equation from it.

(A + B) ^ 2 – (A ^ 2 + B ^ 2) = 422 – 1369.

A ^ 2 + 2AB + B ^ 2 – A ^ 2 – B ^ 2 = 1764 – 1369.

2AB = 395.

Divide both sides of the equation by 4.

½ * AB = 395/4.

S = 98.75.

Thus, we have the area of ​​a right-angled triangle equal to 98.75 cm2.

Answer: the area of ​​a right-angled triangle is 98.75 cm2.