The perimeter of the rectangle is 46 and the diagonal is 227 at the root. find the area of this rectangle.

In order to find the area of ​​a rectangle, we must find the lengths of the sides of the rectangle.

We know that the perimeter of the rectangle is 46 and the diagonal is √227.

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

a + b = 46/2;

a + b = 23.

The sides of the rectangle are the legs of a right-angled 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 = (√227) ^ 2;

a ^ 2 + b ^ 2 = 227;

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;

23 ^ 2 – 227 = 529 – 227 = 302 = 2ab;

ab = 302: 2;

ab = 151.

Answer: 151 sq. units the area of ​​the rectangle.