In the figure at right, $\overleftrightarrow{ E X }$ is tangent to $⊙O$ at point $X$. $OE = 20 \; \text{cm}$ and $XE = 15 \; \text{cm}$.  

1. What is the area of the circle?

   Hint (a):

   According to the definition of the tangent line, what is the measure of $∠EXO$?
   How can you use this to find the length of $XO$?

   Step (a):

   A tangent line is perpendicular to a radius, so the measure of $∠EXO = 90º$, so the triangle is a right triangle. So we can use the Pythagorean Theorem.

   $\begin{array}{l} \left(XE\right)^2 + \left(XO\right)^2 = \left(OE\right)^2 \\ 15^2 + \left(XO\right)^2 = 20^2 \\ 225 + XO^2 = 400 \\ XO^2 = 175 \\ XO=13.23 \end{array}$

   Answer (a):

   $A = \pi r^2\\ A=\pi (13.23)^2\\ A = 175\pi\text{ cm}^2\\ A = 549.78\text{ cm}^2$

2. What is the area of the sector bounded by $\overline{OX}$ and $\overline{ON}$?

   Hint (b):

   Find the measure of $∠XON$. Use this to find the area of the sector.

   Answer (b):

   $A\approx74.21\text{ cm}^2$

3. Find the area of the region bounded by $\overline{XE}$, $\overline{NE}$, and $\overarc{ N X }$.

   Hint (c):

   Find the area of the triangle formed by $E, X,\text{ and }O$. Subtract the area of the sector from the area of the triangle.