A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. What is the length of the fourth side? $\textbf{(A) }200\qquad$ $ \textbf{(B) }200\sqrt{2}\qquad$ $\textbf{(C) }200\sqrt{3}\qquad$ $\textbf{(D) }300\sqrt{2}\qquad$ $\textbf{(E) } 500$
Solution
$ABCD$ is an isosceles trapezoid.
$OE=\sqrt{2\cdot 200^2-100^2}=100\sqrt7$.
Assume $AF=x$. Then $OF=\sqrt{80000-x^2}$ and $EF=\sqrt{40000-(x-100)^2}$, so $OF+EF=OE$.
A quadrilateral $ABCD$ is inscribed in a circle of radius $3\sqrt6$. Three of its sides, $AB$, $BC$ and $CD$, have length $6$. What is the length of the fourth side $DA$?
