An angle bisector of a triangle divides the opposite side of the triangle into segments 6 cm and 8 cm long. A second side of the triangle measures 7 cm long. Find the longest and shortest possible lengths of the third side of the triangle. Round answers to the nearest tenth. (1 point)
Let the third side of the triangle have length $x$. By the angle bisector theorem, the length of the other segment of the opposite side is $\frac{6}{8} \cdot x = \frac{3}{4}x$. Applying the triangle inequality theorem, \begin{align*}
x+7&>\frac{3}{4}x\\
\Rightarrow \qquad \frac{1}{4}x&>7\\
\Rightarrow \qquad x&>28.
\end{align*}Also, \begin{align*}
\frac{3}{4}x+7&>x\\
\Rightarrow \qquad 7&>\frac{1}{4}x\\
\Rightarrow \qquad 28&>x.
\end{align*}Therefore, the length of the third side is between 28 cm and 29 cm, inclusive. The shortest possible length is 28 cm, and the longest possible length is $\boxed{29 \text{ cm}}$.