Question

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

Explanation:

Step1: Find the range of the third - side length

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

Step2: Find the range of the perimeter

The perimeter \(p=a + b + s\), where \(a = 25\), \(b = 35\), and \(10\lt s\lt60\).
Substitute \(a\) and \(b\) into the perimeter formula: \(p=25 + 35+s=60 + s\).
Since \(10\lt s\lt60\), add \(60\) to each part of the inequality.
\(60+10\lt60 + s\lt60 + 60\), so \(70\lt p\lt120\).

Answer:

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