The perimeter of the rectangle is 56 and the diagonal is 27. Find the area of this rectangle.

It is known that the perimeter of the rectangle is 56 and the diagonal is 27.

Let’s find the half-perimeter value of the rectangle:

P = 2 (a + b);

a + b = P / 2;

a + b = 56/2;

a + b = 28.

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

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 = (27) ^ 2;

a ^ 2 + b ^ 2 = 729;

Formula for finding the area of ​​a rectangle:

S = a * b;

Let us express the value ab:

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

282 – 729 = 784 – 729 = 55 = 2ab;

S = ab = 55: 2;

S = ab = 27.5.

Answer: 27.5 sq. units the area of ​​the rectangle.