Question

the lengths of three sides of a triangle are 5, 9, and x, all measured in centimeters. what are all possible values of x?

Explanation:

Step1: Apply triangle inequality theorem

For a triangle with side lengths \(a\), \(b\), and \(c\), the following must hold: \(a + b>c\), \(a + c>b\), and \(b + c>a\). Let \(a = 5\), \(b=9\), and \(c=x\).
First, \(5 + 9>x\), which simplifies to \(x<14\).
Second, \(5+x>9\), which simplifies to \(x > 4\).
Third, \(9 + x>5\) (this is always true for positive \(x\) since \(x>0\) and \(9+0 = 9>5\)).

Answer:

\(4