Pythagorean Theorem (direct and inverse). Isosceles, inscribed, and circumscribed trapezoids.

A circle with radius 6 is circumscribed around trapezoid $ABCD$. The center of this circle lies on the base $AD$, and $BC=4$. Find the area of the trapezoid.
Draw the diagonal of the trapezoid.

## Solution

Let $K$ be the base of the perpendicular dropped from vertex $C$ to $A D$. Find $C K$.

First method. $C K^{2}=A K \cdot K D=1 / 2(A D+B C) \cdot 1 / 2(A D-B C)=8 \cdot 4=32$.

Second method. $C K^{2}=O C^{2}-O K^{2}=6^{2}-(O D-1 / 2 B C)^{2}=6^{2}-2^{2}=32$.

![](https://cdn.mathpix.com/cropped/2024_05_06_14311972f49418f31424g-08.jpg?height=386&width=426&top_left_y=1&top_left_x=822)

$S_{A B C D}=1 / 2(A D+B C) C K=32 \sqrt{2}$.

## Answer

$32 \sqrt{2}$.