The sum of the squares of the sides of the rectangle is 45 cm ^ 2 squared and its perimeter is 18 cm

The sum of the squares of the sides of the rectangle is 45 cm ^ 2 squared and its perimeter is 18 cm. Find the sides of the rectangle.

The solution of the problem.

1. Let’s denote by x one side of the rectangle, and by y the other side of the rectangle.

2. Find the perimeter of the rectangle.

2 * (x + y).

3. Let’s compose and solve the system of equations.

2 * (x + y) = 18;

x ^ 2 + y ^ 2 = 45;

From the first equation of the system, we express y through x.

x + y = 9;

y = 9 – x.

Substitute the value for y in the second equation.

x ^ 2 + (9 – x) ^ 2 = 45;

2x ^ 2 – 18x + 36 = 0;

x ^ 2 – 9x + 18 = 0;

D = 81 – 72 = 9;

The equation has 2 roots x1 = 3 and x2 = 6. Both roots satisfy the condition of the problem.

y1 = 6; y2 = 3. In both cases, one side of the rectangle is 3 cm, and the other 6 cm.

Answer. The sides of the rectangle are 3 cm and 6 cm.