The sides of the rectangle are at 5:12, and the diagonals at the intersection form segments equal to 26.

The sides of the rectangle are at 5:12, and the diagonals at the intersection form segments equal to 26. Find the area of the rectangle.

The intersection point of the diagonals in the rectangle divides them in half. So the diagonal of the rectangle is:

d = 26 * 2 = 52.

The diagonal of a rectangle divides the rectangle into two identical right-angled triangles. We know the ratio of the sides of the triangle (legs) and its hypotenuse. By the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the legs. Let’s solve the problem using the equation:

5x² + 12x² = 52²;

17x² = 2704;

х² = 2704/17;

x = √2704 / 17;

5x = 5 * √2704 / 17 = √13520 / 17 cm – the width of the rectangle;

12x = 12 * √2704/17 = √32448/17 cm – the length of the rectangle;

The area of ​​a rectangle is equal to the product of its width and length:

S = √13520 / 17 * √32448 / 17 = √ (13520/17 * 32448/17) = √ 438696960/289 = √ 1517982 162/289 = 1232 √158 162/289 cm²