# The perimeter of the rectangle is 20cm. Find its sides if you know that the area of the rectangle is 24 cm2.

We introduce variables, let the sides of the rectangle be equal to a and b.

Let us express the perimeter of the rectangle: P = (a + b) * 2. It turns out the equation (a + b) * 2 = 20, hence a + b = 10.

Let us express the area of the rectangle: S = a * b. The equation turns out a * b = 24.

The result is a system of equations:

a + b = 10; a * b = 24. Let us express b from the first equation and substitute it into the second equation.

h = 10 – a.

a (10 – a) = 24;

10a – a ^ 2 – 24 = 0;

-a ^ 2 + 10a – 24 = 0.

Multiply the equation by (-1):

a ^ 2 – 10a + 24 = 0.

We solve the quadratic equation using the discriminant:

a = 1; b = -10; c = 24;

D = b ^ 2 – 4ac; D = (-10) ^ 2 – 4 * 1 * 24 = 100 – 96 = 4 (D = 2);

x = (-b ± √D) / 2a;

a1 = (10 – 2) / 2 = 8/2 = 4.

a2 = (10 + 2) / 2 = 12/2 = 6.

Let’s calculate the value in:

h = 10 – a.

1) a = 4; h = 10 – 4 = 6.

2) a = 6; h = 10 – 6 = 4.

Answer: the sides of the rectangle are 4 and 6 cm.