The perimeter of the quadrilateral 34 cm diagonal divides the quadrilateral into two triangles whose perimeters

The perimeter of the quadrilateral 34 cm diagonal divides the quadrilateral into two triangles whose perimeters are 20 cm and 26 cm Find the length of the diagonal.

Let us denote the lengths of the sides of this quadrangle by x1, x2, x3 and x4, and the length of the diagonal by y.
According to the condition of the problem, the diagonal divides the quadrangle into two triangles with sides x1, x2, y and x3, x4, y, the perimeters of which are 20 cm and 26 cm, therefore, the following relations hold:
x1 + x2 + y = 20;
x3 + x4 + y = 26.
Adding these two ratios, we get:
x1 + x2 + y + x3 + x4 + y = 20 + 26;
x1 + x2 + x3 + x4 + 2 * y = 46.
According to the condition of the problem, the perimeter of the quadrangle is 34 cm, therefore, the following relation holds:
x1 + x2 + x3 + x4 = 34.
Substituting this value for x1 + x2 + x3 + x4 into the ratio x1 + x2 + x3 + x4 + 2 * y = 46, we get:
34 + 2 * y = 46.
We solve the resulting equation:
2 * y = 46 – 34;
2 * y = 12;
y = 12/2;
y = 6 cm.

Answer: the length of the diagonal is 6 cm.