The area of the rectangle is 1080 cm2, and its perimeter is 138 cm. Find the length of the diagonal and sides of the rectangle.

As you know, the perimeter of a rectangle is calculated by the formula: P = (a + b) * 2, where a is the length of the rectangle, b is the width of the rectangle.
Substitute all known values ​​in the formula: 138 = (a + b) * 2.
As you know, the area of ​​a rectangle is calculated by the formula: S = a * b, where a is the length of the rectangle, b is the width of the rectangle.
Substitute all known values ​​in the formula: 1080 = a * b.
a = 1080 / b.
Substitute instead of a into the perimeter formula and find side b: (1080 / b + b) * 2 = 138.
2160 / b + 2b = 138.
2160 + 2b² = 138b.
2b² – 138b + 2160 = 0.
D = b² – 4ac = 19,044 – 4 * 2 * 2160 = 19,044 – 17,280 = 1,764.
x1 = (-b + √D) / 2a = (138 + 42) / 2 * 2 = 180/4 = 45.
x2 = (-b – √D) / 2a = (138 – 42) / 2 * 2 = 96/4 = 24.
Side b is either 45 cm or 24 cm.
Find the side a = 1080/45 = 24 cm.

Answer: The first side of the rectangle is 24 cm, the second side is 45 cm.
Find the diagonal from the Pythagorean theorem: c² = a² + b².
c² = 24² + 45².
c² = 576 + 2,025 = 2,601.
c = √2 061 = 51.
Answer: the diagonal is 51 cm.