Question

here is triangle abc. write an inequality to show all possible side lengths for side ab.

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. Let the sides of the triangle be \(a = 5\), \(b\) (side \(AB\)), and \(c = 17\).

Step2: First inequality

\(a + b>c\), so \(5 + b>17\), which simplifies to \(b>17 - 5\), or \(b > 12\).

Step3: Second inequality

\(a + c>b\), so \(5+17>b\), or \(b<22\).

Step4: Third inequality

\(b + c>a\), which is always true for non - negative values of \(b\) since \(c = 17\) and \(a = 5\) (\(b+17>5\) for \(b>0\)).

Answer:

\(12 < AB<22\)