The perimeter of the rectangle is 13 cm. And its area is 10.5 cm2, find the sides of the rectangle.

1. Let u and v be the sides of the rectangle. Then:

{2 (u + v) = 13;
{uv = 10.5.

2. Let’s open the brackets and multiply the second equation by 4:

{2u + 2v = 13;
{4uv = 42;
{2u + 2v = 13;
{(2u) * (2v) = 42.
3. By the converse Vieta theorem, 2u and 2v are the roots of the equation:

t ^ 2 – 13t + 42 = 0;
D = 13 ^ 2 – 4 * 42 = 169 – 168 = 1;
t = (13 ± 1) / 2;
t1 = (13 – 1) / 2 = 12/2 = 6;
t2 = (13 + 1) / 2 = 14/2 = 7.
4. Sides of the rectangle:

a)

{2u = 6;
{2v = 7;
{u = 3;
{v = 3.5.
b)

{2u = 7;
{2v = 6;
{u = 3.5;
{v = 3.
Answer: 3 cm and 3.5 cm.