Dissecting Triangles

We are asked to find the length of the segment $\overline{AC}.$ Let's consider the diagram below.

As we can see, the segment $\overline{AC}$ consists of two segments: $\overline{AD}$ and $\overline{DC}.$ Thus, by the Segment Addition Postulate, its length equals the sum of these segments lengths. $$\begin{array}{r}AC=AD+DC\end{array}$$ The measure $AD$ is known from the diagram. How can find the measure of $DC?$ From the diagram, we see that segments $\overline{AB}$ and $\overline{BC}$ have the same length. This means, point $B$ is equidistant form $A$ and $C.$ Let's now use the Converse of the Perpendicular Bisector Theorem. $$\begin{array}{c}\text{If a point is equidistant }\\ \text{from the endpoints of a segment,}\\ \text{then it is on the perpendicular bisector }\\ \text{ of the segment.}\end{array}$$ According to the theorem, $\overline{BD}$ is a perpendicular bisector of the segment $AC.$ Therefore, $D$ is a midpoint of $\overline{AC}.$ We conclude that the measure of $\overline{DC}$ is also $3.5.$ Finally, we can substitute $AD$ with $3.5$ and $DC$ with $3.5$ in the above equation, and calculate $AC.$ $$\begin{array}{r}AC=3.5+3.5=7\end{array}$$