Given: trapezoid – ABCD, angle A = 90 degrees, C = 135 degrees, AB = 5 cm, AD = 20.5 cm Find: the middle line.

From the top of the trapezoid, we lower the height CH. In a right-angled triangle СНD, we define the value of the angle НСD.

Angle НСD = ВСD – ВСН = 135 – 90 = 45.

Then the angle СDН = 180 – 90 – 45 = 45.

Then the triangle CHD is isosceles, since the angles at the base of CD are equal, and therefore CH = DH.

Since the quadrangle ABCH is a rectangle, then AH = BC, and CD = AB = 5 cm, which means DH = CH = 5 cm.

Then the length of the segment AH = AD – DH = 20.5 – 5 = 15.5 cm, then BC = 15.5 cm.

Let’s determine the value of the middle line of the KR.

KR = (AD + BC) / 2 = (20.5 + 15.5) / 2 = 18 cm.

Answer: The length of the middle line is 18 cm.