3. In an isosceles trapezoid with an area of 20, a circle with a radius of 2 is inscribed. Determine the length of each side of the trapezoid.
Solution. The diameter of the inscribed circle is equal to the height $h$ of the given trapezoid. Thus, $h=4$. For the area of the trapezoid, we have $P=\frac{a+b}{2} h=20$, from which we get $a+b=10$. Since the given trapezoid is tangential, the sums of the lengths of its opposite sides are equal, so $a+b=2 c$, from which we get $c=5$. From the Pythagorean theorem, we get

![](https://cdn.mathpix.com/cropped/2024_06_05_b87224e82046fedb440bg-2.jpg?height=387&width=471&top_left_y=345&top_left_x=1009)
that: $\frac{a-b}{2}=\sqrt{5^{2}-4^{2}}=3$. Finally, solving the system of equations

$$
\left\{\begin{array}{l}
a+b=10 \\
a-b=6
\end{array}\right.
$$

we get $a=8$ and $b=2$.