Question

given a triangle that has sides of 9 and 16. what is the range of the perimeter?

Explanation:

Step1: Recall triangle - side inequality

Let the sides of the triangle be \(a = 9\), \(b = 16\), and \(c\) be the third - side. According to the triangle - side inequality theorem, \(|a - b|\lt c\lt a + b\).
So, \(|9 - 16|\lt c\lt9 + 16\), which simplifies to \(7\lt c\lt25\).

Step2: Calculate the perimeter formula

The perimeter \(P\) of a triangle with sides \(a\), \(b\), and \(c\) is \(P=a + b + c\). Substituting \(a = 9\) and \(b = 16\), we get \(P=9 + 16 + c=25 + c\).

Step3: Find the range of the perimeter

Since \(7\lt c\lt25\), we add 25 to each part of the inequality.
\(25+7\lt25 + c\lt25 + 25\).
So, \(32\lt P\lt50\).

Answer:

The range of the perimeter is \(32\lt P\lt50\).