A machine-shop cutting tool has the shape of a notched circle, as shown. The radius of the circle is $\sqrt{50}$ cm, the length of $AB$ is $6$ cm and that of $BC$ is $2$ cm. The angle $ABC$ is a right angle. Find the square of the distance (in centimeters) from $B$ to the center of the circle.
Because we are given a right angle, we look for ways to apply the [Pythagorean Theorem](https://artofproblemsolving.com/wiki/index.php/Pythagorean_Theorem).  Let the foot of the [perpendicular](https://artofproblemsolving.com/wiki/index.php/Perpendicular) from $O$ to $AB$ be $D$ and let the foot of the perpendicular from $O$ to the [line](https://artofproblemsolving.com/wiki/index.php/Line) $BC$ be $E$. Let $OE=x$ and $OD=y$. We're trying to find $x^2+y^2$.


Applying the Pythagorean Theorem, $OA^2 = OD^2 + AD^2$ and $OC^2 = EC^2 + EO^2$.
Thus, $\left(\sqrt{50}\right)^2 = y^2 + (6-x)^2$, and $\left(\sqrt{50}\right)^2 = x^2 + (y+2)^2$. We solve this system to get $x = 1$ and $y = 5$, such that the answer is $1^2 + 5^2 = \boxed{026}$.