Question

applying the triangle inequality theorem
in triangle abc, ab measures 25 cm and ac measures 35 cm.
the inequality < s < represents the possible third side length of the triangle, s, in centimeters
the inequality < p < represents the possible values for the perimeter, p, of the triangle, in centimeters

Explanation:

Step1: Find the range of the third side \(s\)

By the triangle - inequality theorem, for a triangle with side lengths \(a\), \(b\), and \(c\), \(|a - b|\lt c\lt a + b\).
Here \(a = 25\) and \(b = 35\).
\(|25-35|=| - 10| = 10\) and \(25 + 35=60\). So \(10\lt s\lt60\).

Step2: Find the range of the perimeter \(p\)

The perimeter \(p=a + b + s\), where \(a = 25\), \(b = 35\), and \(10\lt s\lt60\).
Substitute \(s\) into the perimeter formula:
\(p=(25 + 35)+s=60 + s\).
When \(s = 10\), \(p=60 + 10=70\); when \(s = 60\), \(p=60+60 = 120\). So \(70\lt p\lt120\).

Answer:

The first inequality is \(10\lt s\lt60\) and the second inequality is \(70\lt p\lt120\).