The perimeter of the rectangle is 74 and the diagonal is 36 find the area of this rectangle

We need to find the area of ​​the rectangle.

It is known that the perimeter of the rectangle is 74 and the diagonal is 36.

Find the half-perimeter of the rectangle: P = 2 (a + b); a + b = P / 2;

a + b = 74/2;

a + b = 37.

The sides of the rectangle are the legs of the right triangle, and the diagonal of the rectangle is the hypotenuse.

To find the hypotenuse, we will use the Pythagorean theorem.

The sum of the squares of the legs is equal to the square of the hypotenuse.

a ^ 2 + b ^ 2 = c ^ 2;

a ^ 2 + b ^ 2 = (36) ^ 2;

a ^ 2 + b ^ 2 = 1296;

Formula for finding the area of ​​a rectangle:

S = a * b;

Let’s express the value of ab:

(a + b) ^ 2 – (a ^ 2 + b ^ 2) = a ^ 2 + 2ab + b ^ 2 – a ^ 2 – b ^ 2 = 2ab;

372 – 1296 = 1369 – 1296 = 73 = 2ab;

ab = 73: 2;

ab = 36.5.

Answer: 36.5 sq. units the area of ​​the rectangle.