Question

a triangle has two sides of length 1 and 10. 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 inequalities \(|a - b|\lt x\lt a + b\). Here \(a = 1\) and \(b = 10\).

Step2: Calculate the lower - bound

First, calculate \(|a - b|=|1 - 10|=9\). The inequality for the third - side length \(x\) is \(9\lt x\lt11\).

Step3: Find the smallest whole - number

Since \(x\) is a whole number and \(9\lt x\), the smallest whole - number value of \(x\) is \(10\).

Answer:

10