Question

a triangle has two sides of length 4 and 19. what is the smallest possible whole - number length for the third side?

Explanation:

Step1: Recall triangle - inequality theorem

Let the sides of the triangle be \(a = 4\), \(b = 19\), and \(c\) be the third - side. The triangle - inequality theorem states that \(|a - b|\lt c\lt a + b\).

Step2: Calculate the lower - bound

First, calculate \(|a - b|\). Since \(|4 - 19|=| - 15| = 15\) and \(a + b=4 + 19 = 23\), the inequality for the third - side is \(15\lt c\lt23\).

Step3: Find the smallest whole - number

The smallest whole - number greater than 15 is 16.

Answer:

16