Dissecting Triangles

We want to find the value of $x$ in the given diagram.

We know that $\overline{DB}$ is the perpendicular bisector of $\overline{AC}.$ The Perpendicular Bisector Theorem tells us that, if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Because $D$ lies on $\overline{DB},$ it is equidistant from the endpoints of $\overline{AC}.$ $$\begin{array}{c}AD=DC\Rightarrow 9x-18=3x\end{array}$$ To find the value of $x,$ we can solve the above equation.