Described quadrilaterals Area of a trapezoid

The bases of an isosceles trapezoid circumscribed around a circle are 2 and 18. Find the area of the trapezoid.

#
Let $C H$ be the height of the given isosceles trapezoid $A B C D$ with bases $A D=18$ and $B C=2$. Then

$$
D H=\frac{1}{2}(A D-B C)=\frac{1}{2}(18-2)=8
$$

Since the trapezoid is circumscribed around a circle, the sums of its opposite sides are equal, meaning the sum of the lateral sides is equal to the sum of the bases, i.e., 20. Since the trapezoid is isosceles, $C D=10$. From the right triangle $C D H$, we find that

$$
C H=\sqrt{C D^{2}-D H^{2}}=\sqrt{100-64}=6
$$

$$
\begin{gathered}
\text { Therefore, } \\
S_{A B C D}=\frac{1}{2}(A D+B C) \cdot C H=\frac{1}{2}(18+2) \cdot 6=60 .
\end{gathered}
$$

## Answer

60.00

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