10.231. A circle of radius $r$ is inscribed in a rectangular trapezoid. Find the sides of the trapezoid if its smaller base is equal to $4 r / 3$.
## Solution.

Let's find the side $A B=2 r$ (Fig. 10.39). Let $E D=x$. Then, using the equality $A B+C D=B C+A D$, we get $2 r+C D=\frac{4 r}{3}+\frac{4 r}{3}+x$, from which $C D=\frac{2 r}{3}+x$. In $\triangle C E D$ we have $C D^{2}=C E^{2}+E D^{2} \Leftrightarrow\left(\frac{2 r}{3}+x\right)^{2}=4 r^{2}+x^{2}$, from which $x=\frac{8 r}{3}$. We obtained $C D=\frac{10 r}{3}, A D=\frac{12 r}{3}=4 r$.

Answer: $4 r, \frac{10 r}{3}, 2 r$.