# The perimeter of the rectangle is 26 cm and its area is 36 cm2. find the lengths of the sides of the rectangle.

1) The area of an arbitrary rectangle: S = a * b, where S is the area (S = 36 cm2), a is the length of the larger side, b is the length of the smaller side.

2) The perimeter of an arbitrary rectangle: P = 2a + 2b, where P is the perimeter (P = 26 cm).

3) Let us express the length of the rectangle: a = (P – 2b) / 2.

4) Determine the lengths of the sides: S = (P – 2b) / 2 * b.

S / b = (P – 2b) / 2 | * b.

S = 0.5P * b – b ^ 2.

b ^ 2 – 0.5P * b + S = 0.

The resulting quadratic equation: b ^ 2 – 0.5 * 26 * b + 36 = 0.

b ^ 2 – 13 * b + 36 = 0.

By Vieta’s theorem:

b1 + b2 = 13 and b1 * b2 = 36, whence b1 = 9 cm and b2 = 4 cm (both roots of the equation are correct, hence the lengths of the sides: a = 9 cm, b = 4 cm).

Answer: The length of the larger side is 9 cm, the smaller side is 4 cm.