Question

a triangle has two sides of length 11 and 3. what is the largest possible whole - number length for the third side?

Explanation:

Step1: Recall triangle - inequality theorem

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Let the sides of the triangle be \(a = 11\), \(b = 3\), and \(c\) be the third - side. So \(a + b>c\), \(a + c>b\), and \(b + c>a\). Also, \(|a - b|

Step2: Find the upper - bound for the third side

From \(a + b>c\), we have \(11 + 3>c\), i.e., \(c<14\). From \(|a - b|

Step3: Determine the largest whole - number value

Since \(8 < c<14\), the largest whole - number value of \(c\) is 13.

Answer:

13