Question

a triangle has two sides of length 8 and 19. what is the smallest possible whole - number length for the third side?

Explanation:

Step1: Recall triangle - inequality theorem

The length of the third side \(x\) of a triangle with side lengths \(a\) and \(b\) satisfies the inequality \(|a - b|\lt x\lt a + b\). Here \(a = 8\) and \(b = 19\).

Step2: Calculate the lower - bound

First, calculate \(|19 - 8|=11\). Since \(x\gt|19 - 8|\), and \(x\) is a whole number, the smallest whole - number value of \(x\) that satisfies this inequality is \(12\).

Answer:

12