Two non-congruent triangles have the same area. Each triangle has sides of length $7$ and $9$, and the third side of each triangle has integer length. What is the sum of the lengths of the third sides?

1. Let $\gamma$ be the angle between the sides $7$ and $9$. The area is $\frac12\cdot7\cdot9\sin\gamma$, so two non-congruent triangles of equal area have angles $\gamma$ and $180^\circ-\gamma$.
2. By the law of cosines, the third sides satisfy $c^2=130-126\cos\gamma$ and $c'^2=130+126\cos\gamma$, so $c^2+c'^2=260$.
3. Since $2
4. So the third sides are $8$ and $14$. Both triangles exist ($7+9>14$), and their sum is $\boxed{22}$.