The diagonal divides a quadrilateral with a perimeter of 31 cm into two triangles with perimeters

The diagonal divides a quadrilateral with a perimeter of 31 cm into two triangles with perimeters of 21 cm and 30. Determine the length of this diagonal.

Let AВСD be the given quadrangle, AC the diagonal.

P (AВСD) = 31 cm; P (ABC) = 21 cm; P (ADС) = 30 cm.

Let’s write down all the perimeters:

P (AВСD) = AB + BC +СD + AD.

P (ABC) = AB + BC + AC.

P (ADС) = AD + СD + AC.

Let us express the sum of the perimeters of the triangles:

P (ABC) + P (ADС) = AB + BC + AC + AD +СD + AC = (AB + BC +СD + AD) + 2 * AC.

Since P (ABC) + P (ADС) = 21 + 30 = 51. A (AB + BC + СD + AD) = P (AВСD) = 31, the equation is obtained:

31 + 2 * AC = 51.

2 * AC = 51 – 31.

2 * AC = 20.

AC = 10 (cm).

Answer: the diagonal is 10 cm.