Points \(A(7,12)\), \(B(3,2)\), \(C(11,2)\), \(D(6,2)\) and \(E(10,2)\) are plotted on the Cartesian plane. The point \(F(8,k)\) lies inside \(\triangle ABC\) so that the area inside of \(\triangle ABC\) but outside of \(\triangle DEF\) is equal to \(32\text{ units}^2\).
Determine the value of \(k\), the \(y\)-coordinate of \(F\).
Themes: Algebra, Geometry & Measurement