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)

Answer 1

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 $\boxed{29 \text{ cm}}$.