1. The area of the circle that passes through the points $(1,1),(1,7)$, and $(9,1)$ can be expressed as $k \pi$. What is the value of $k$ ?
Solution: Consider the triangle formed by the three given points. The triangle has two sides parallel to the axes, so it is right angled. The sides parallel to the axes have lengths $7-1=6$ and $9-1=8$. By the Pythagorean Theorem, the length of the hypotenuse is $\sqrt{6^{2}+8^{2}}=10$.
Since the triangle formed by the three points is right angled, the hypotenuse is the diameter of the circle passing through those points. Thus, the area of the circle is $5^{2} \pi=25 \pi$ so $k=25$.