Question

what is the range for the possible side length \\( \overline { a b } \\) of \\( \triangle a b c \\)?
\\( \bigcirc 2 < a b < 18 \\)
\\( \bigcirc 2 > \overline { a b } > 18 \\)
\\( \bigcirc 2 > \overline { a b } > 10 \\)
\\( \bigcirc 2 < a b < 10 \\)

Explanation:

Step1: Apply triangle inequality theorem

The triangle inequality theorem states that for any triangle with sides \(a\), \(b\), and \(c\), \(|a - b|\lt c\lt a + b\). Let \(a = 10\), \(b=8\), and \(c=x\) (where \(x\) represents the length of \(\overline{AB}\)).
First, find the lower - bound: \(|10 - 8|\lt x\), which simplifies to \(2\lt x\).

Step2: Find the upper - bound

Then, find the upper - bound: \(x\lt10 + 8\), which simplifies to \(x\lt18\).

Answer:

\(2\lt AB\lt18\) (the first option)