Question

Describe the possible lengths of the third side of a triangle if the two given sides are 6 inches and 9 inches long.

Answer 1

Recall that for a, b, and c to be the lengths of a triangle they must satisfy the following inequalities:

$\begin{gathered} a+b>c, \\ a+c>b, \\ b+c>a\text{.} \end{gathered}$

Therefore, if we call the other side x, it must satisfy that:

$\begin{gathered} 6+x>9, \\ 9+x>6, \\ 9+6>x\text{.} \end{gathered}$

Solving each inequality for x, we get:

$\begin{gathered} x>9-6=3, \\ x>6-9=-3, \\ 15>x\text{.} \end{gathered}$

Therefore, x must be greater than 3 inches but less than 15 inches.

Answer: The possible lengths of the third side must be greater than 3 inches but less than 15 inches