One of the corners of a rectangular trapezoid is 135 degrees, the middle line is 21 cm, and the bases are 5: 2.

One of the corners of a rectangular trapezoid is 135 degrees, the middle line is 21 cm, and the bases are 5: 2. Find the smaller side of the trapezoid.

The length of the center line is calculated using the formula:
AB + CD = 2 * EF = 42, hence AB = 42 – CD.
By the condition of the problem:
2 * AB = 5 * CD
2 * (42 – CD) = 5 * CD
CD = 42 * 2/7 = 12 cm
AB = 42 – 12 = 30 cm.
Let DG be the height drawn to the base AB.
GB = DC; AG = AB – GB = 30 – 12 = 18. DG = CB.
Consider triangle AGD.
Angle AGD = 90, angle ADG = angle ADC – angle GDC = 135 – 90 = 45, angle DAC = 180 – 90 – 45 = 45.
Triangle ADG is isosceles with base AD, therefore DG = AG = 18.
CB = 18.
Answer: 18 cm.