In the coordinate plane, let point A be located at $(0, 0)$ and point B at $(8, 6)$. A point C lies in the first quadrant such that the angle $ heta$ between line segments AC and BC is $30^ ext{o}$. Given the angle $ heta$ and the lengths of segments AC and BC, calculate the length of segment AC. Use the Law of Cosines, which states that for any triangle with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively, the following formula holds:

$$ c^2 = a^2 + b^2 - 2ab imes ext{cos}(C) $$

Let AC be $c$, BC be $b$, and the distance AB be $d = 10$ units (determined by the distance formula). As point C lies in the first quadrant, find the value of $AC$.