A circle is inscribed in the parallelogram. Find the perimeter of the parallelogram if one of its sides is 6.

In order to inscribe a circle into a quadrangle, and in our case a parallelogram, the following conditions must be met:

– the described quadrilateral (parallelogram) must be convex;

– the sums of its opposite sides must be equal: AB + CD = BC + AD.

We use the properties of the parallelogram – the opposite sides of the parallelogram are equal, that is:

AB = CD = x;

BC = AD = y.

We compose and solve the equation.

x + x = y + y;

2 * x = 2 * y;

x = y.

The resulting expression says that all sides of our parallelogram are equal, therefore, this is a rhombus.

Find the perimeter of the rhombus. The perimeter of a rhombus is equal to the sum of four lengths of its sides or the product of the length of any of its sides by four (since the lengths of all sides of a rhombus are equal).

The perimeter of the rhombus is:

P = 6 * 4 = 24.

Answer: the perimeter of the parallelogram is 24.