A triangle with vertices $(3,4)$, $(5,-3)$, and $(7,2)$ is reflected about the line $x=5$ to create a second triangle. What is the area of the union of the two triangles?

1. Each triangle has area $12$ (by the shoelace formula 15).
2. By symmetry, the two triangles overlap in a kite with vertical diagonal from $(5,3)$ to $(5,-3)$ (length $6$) and horizontal diagonal from $(3.5,2.25)$ to $(6.5,2.25)$ (length $3$), so the overlap area is $\frac12(6)(3)=9$.
3. Union area $=2(12)-9=\boxed{15}$.