The perimeter of the rectangle is 34cm. Find the sides of the rectangle if the diagonal is 13cm long.

Let’s say that the sides of the rectangle are equal to x and y centimeters.

Since the perimeter of the rectangle is 34, we get:

2 * (x + y) = 34,

x + y = 17,

y = 17 – x.

The diagonal of a rectangle is the hypotenuse of a right-angled triangle, the legs of which are equal to the sides of the rectangle.

By the Pythagorean theorem we get:

x² + y² = 13²,

x² + (17 – x) ² = 169,

x² + 289 – 34 * x + x² = 169,

2 * x² – 34 * x + 120 = 0.

The discriminant of this equation is:

(-34) ² – 4 * 2 * 120 = 196.

The problem has the following solutions:

x = (34 + 14) / 4 = 12 and x = (34 – 14) / 4 = 5.

If one side is 12 cm, then the other side is 17 – 12 = 5 cm.

If one side is 5 cm, then the other is 17 – 5 = 12 cm.

Answer: 12 cm and 5 cm.