Since the trapezoid is at same time tangent quadrilateral, it can be written:

$$a+b=2c$$

$$A=s·r=\frac{\left(a+b+2c\right)}{2}·r$$

$$\sin 30°=\frac{h}{c}$$

$$\frac{1}{2}=\frac{h}{c}$$

$$c=2·h$$

$$h=2·r$$

$$c=2·h=2·2·r=4r$$

$$r=\frac{c}{4}$$

$$A=\frac{\left(a+b+2c\right)}{2}·r=\frac{\left(2c+2c\right)}{2}·\frac{c}{4}=\frac{\bcancel{4}c}{2}·\frac{c}{\bcancel{4}}=\frac{c^2}{2}$$

$$50=\frac{c^2}{2}$$

$$100=c^2$$

$$c=\sqrt{100}$$