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

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 \(s\) is the third side.
Since \(10\lt s\lt60\), we substitute \(s\) into the perimeter formula.
\(p=25 + 35+s=60 + s\).
When \(s = 10\), \(p=60+10 = 70\) (but \(s>10\)), when \(s = 60\), \(p=60 + 60=120\) (but \(s<60\)). So \(70\lt p\lt120\).

Answer:

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