The rectangle has a perimeter of 26 cm and an area of 42 cm2. 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. The perimeter of the rectangle is 2 * (x + y).
3. The area of the rectangle is xy.
4. Let’s compose and solve the system of equations.
2 * (x + y) = 26;
xy = 42;
From the first equation of the system, we express y through x.
x + y = 13;
y = 13 – x;
Substitute the value for y in the second equation.
x * (13 – x) = 42;
x ^ 2 – 13x + 42 = 0;
D = 169 – 168 = 1;
The equation has 2 roots x1 = 6 and x2 = 7.
Both roots satisfy the condition of the problem.
5. Find y.
y = 13 – x;
y1 = 13 – 6 = 7;
y2 = 13 – 7 = 6;
6. In both cases, one side of the rectangle is 6 cm and the other 7 cm.
Answer. The lengths of the sides of the rectangle are 6 cm, 7 cm.