Question

given the side lengths of 6 and 16, determine the range of the perimeter.

Explanation:

Step1: Apply triangle - side - length rule

In a triangle, the length of any side must be greater than the difference and less than the sum of the lengths of the other two sides. Let the sides of the triangle be \(a = 6\), \(b = 16\), and \(c=x\). Then \(|a - b|\lt x\lt a + b\). Calculate \(|6 - 16|=10\) and \(6 + 16 = 22\), so \(10\lt x\lt22\).

Step2: Calculate the perimeter formula

The perimeter \(P\) of a triangle with side - lengths \(a\), \(b\), and \(c\) is \(P=a + b + c\). Substitute \(a = 6\) and \(b = 16\) into the perimeter formula: \(P=6 + 16+x=22 + x\).

Step3: Find the range of the perimeter

Since \(10\lt x\lt22\), add 22 to each part of the inequality. \(22+10\lt22 + x\lt22 + 22\). So \(32\lt P\lt44\).

Answer:

The range of the perimeter is \(32\lt P\lt44\).