Question

if a triangle has lengths of 3 ft and 54 ft, check all the possible lengths for the third side.
□ 51 ft
□ 53 ft
□ 57 ft
□ 55 ft
□ 58 ft

Explanation:

Step1: Determine the range of the third side

According to the triangle inequality theorem, for a triangle with side lengths \(a\), \(b\), and \(c\), \(|a - b|\lt c\lt a + b\). Here \(a = 3\) and \(b=54\).
First, calculate \(|a - b|\): \(|3 - 54|=51\).
Then, calculate \(a + b\): \(3+54 = 57\).
So, \(51\lt c\lt57\).

Step2: Check each option

• For \(51\) ft: Since \(c>51\) (not \(c\geq51\)), \(51\) ft is not valid.
• For \(53\) ft: Since \(51\lt53\lt57\), \(53\) ft is valid.
• For \(57\) ft: Since \(c<57\) (not \(c\leq57\)), \(57\) ft is not valid.
• For \(55\) ft: Since \(51\lt55\lt57\), \(55\) ft is valid.
• For \(58\) ft: Since \(58>57\), \(58\) ft is not valid.

Answer:

\(53\) ft, \(55\) ft