Question

determine the possible range of the third side of the triangle, given the side lengths of 9 and 9.

Explanation:

Step1: Recall triangle - inequality theorem

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side, and the difference between the lengths of any two sides must be less than the length of the third side. Let the two given side - lengths be \(a = 9\) and \(b = 9\), and the third side be \(x\).

Step2: Calculate the lower - bound

The difference between the two sides gives the lower - bound of the third side. \(|a - b|\lt x\). Since \(a=b = 9\), \(|9 - 9|=0\), so \(x>0\).

Step3: Calculate the upper - bound

The sum of the two sides gives the upper - bound of the third side. \(x\lt a + b\). Substituting \(a = 9\) and \(b = 9\), we get \(x\lt9 + 9=18\).

Answer:

\(0\lt x\lt18\)