Find the length of the diagonal of a rectangle with an area of 480 cm² and a perimeter of 92 cm.

The area of ​​a rectangle is equal to the product of two adjacent sides:

S = a * b = 480.

The perimeter is the sum of the lengths of all sides, and since the opposite sides in a rectangle are equal, the sum of the lengths of two adjacent sides is equal to half the perimeter:

a + b = P / 2;

a + b = 46.

We square both sides of this equality, we get:

(a + b) ^ 2 = 46 ^ 2;

a ^ 2 + b ^ 2 + 2 * a * b = 2116;

a ^ 2 + b ^ 2 = 2116 – 2 * a * b = 2116 – 2 * S = 2116 – 2 * 480 = 1156.

The adjacent sides of the rectangle and its diagonal form a right-angled triangle in which the diagonal is the hypotenuse, therefore:

d ^ 2 = a ^ 2 + b ^ 2 = 1156;

d = √1156 = 34 cm – the diagonal of the rectangle.