Let the base $AB$ of the triangle be the side of length $8\textrm{ cm}$ and let $AC$ be the side of length $6\textrm{ cm}$. So $C$ must lie on the circle with centre $A$ and radius $6\textrm{ cm}$ as shown. The area of the triangle is to be $7\textrm{ cm}^2$, so the perpendicular from $C$ to $AB$ (or to $BA$ produced) must be of length $\frac{7}{4}\textrm{cm}$.
The diagram shows the four possible positions $D$, $E$, $F$ and $G$ of $C$. However, since $∠BAG = ∠BAF$ and $∠BAD = ∠BAE$, these correspond to exactly two possibilities for the length of the third side $AC$. The diagrams below show the two possibilities.