A circle is inscribed in an isosceles trapezoid. The middle line is 4 cm. Angle 30 degrees. Find the radius of the circle.

Let’s apply the formula for the length of the midline of a trapezoid.
KP = (BC + AD) / 2 = 4 cm.
(BC + AD) = 2 * 4 = 8 cm.
Since a circle is inscribed in the trapezoid, the sum of the lengths of its lateral moans is equal to the sum of the lengths of its bases. Then (AB + CD) = (BC + AD) = 8 cm.
Then AB = BC = 8/2 = 4 cm.
Let’s draw the height of the HB trapezoid. In a right-angled triangle BAH, the BH leg is cut against an angle of 300, then BH = AB / 2 = 4/2 = 2 cm.
The radius of a circle inscribed in a trapezoid is half the length of its height.
R = ОМ = ВН / 2 = 2/2 = 1 cm.
Answer: The radius of the circle is 1 cm.