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: Recall triangle - inequality theorem

The length of the third side \(s\) of a triangle with side lengths \(a\) and \(b\) satisfies \(|a - b|\lt s\lt a + b\). Here \(a = 25\) and \(b = 35\).

Step2: Calculate the range of the third - side length

\(|35 - 25|=10\) and \(35 + 25 = 60\). So \(10\lt s\lt60\).

Step3: Calculate the perimeter formula

The perimeter \(p\) of \(\triangle ABC\) is \(p=AB + AC+s=25 + 35+s=60 + s\).

Step4: Find the range of the perimeter

Since \(10\lt s\lt60\), add 60 to each part of the inequality. \(10+60\lt60 + s\lt60 + 60\), so \(70\lt p\lt120\).

Answer:

The first inequality: \(10\lt s\lt60\)
The second inequality: \(70\lt p\lt120\)