A diagonal divides a quadrangle with a perimeter of 31 cm into two triangles with a perimeter

A diagonal divides a quadrangle with a perimeter of 31 cm into two triangles with a perimeter of 21 cm and 30 cm. Determine the length of this diagonal.

Perimeter of a quadrilateral with sides a, b, c, d and diagonal e:

Pabcd = a + b + c + e = 31.

Perimeters of triangles:

Pabf = a + b + e;

Pcdf = c + d + e;

Pabf + Pcdf = a + b + e + c + d + e = (a + b + c + d) + 2e.

The expression in parentheses equals the perimeter of Pabcd:

Pabf + Pcdf = Pabcd + 2e.

The last equality could be written immediately, proceeding from the fact that the sum of the perimeters of two triangles is more than the perimeter of the quadrangle by two diagonals e.

21 cm + 30 cm = 31 cm + 2e;

e = (21 cm + 30 cm – 31 cm) / 2 = 10 cm.

Answer: 10 cm.