Question

a triangle has two sides of length 8 and 12. what is the smallest possible whole - number length for the third side?

Explanation:

Step1: Recall triangle - inequality theorem

The length of the third side \(c\) of a triangle with side lengths \(a\) and \(b\) satisfies the inequality \(|a - b|\lt c\lt a + b\). Here \(a = 8\) and \(b = 12\).

Step2: Calculate the lower - bound

First, find \(|a - b|\), where \(|8 - 12|=| - 4| = 4\). Since \(c\) is a whole - number and \(c>4\), the smallest whole - number value for \(c\) is \(5\).

Answer:

5