A circle is inscribed in a rectangular trapezoid with a perimeter of 40.

A circle is inscribed in a rectangular trapezoid with a perimeter of 40. The large side of the trapezoid is 11. Find the radius of the circle.

Let a trapezoid ABCD be given, r is the radius of the circle.
The points of tangency of the sides with the circle divide the sides into two parts: AB at point M, BC at point K, CD at point F, AD at point E.
By the property of a trapezoid with an inscribed circle, sides
AB = AM + BM = r + r;
BC = BK + KC = r + x;
CD = CF + FD = x + y = 11 cm, y = 11 – x;
AD = AE + DE = r + y = r + 11 – x.
Let’s write down the perimeter formula:
AB + BC + CD + AD = r + r + r + x + 11 + r + 11 – x = 40;
4 * r + 22 = 40;
4 * r = 18;
r = 4.5 (cm).
The radius of the circle is 4.5 cm.