Question

two sides of a triangle measure 12 and 10. which inequality shows all the possible lengths of the third side, x? a 2 < x < 22 b 10 < x < 12 c 0 < x < 22 d 3 < x < 21

Explanation:

Step1: Recall triangle - side rule

The length of the third side \(x\) of a triangle with side lengths \(a\) and \(b\) must satisfy the following two inequalities: \(|a - b|\lt x\) and \(x\lt a + b\).

Step2: Calculate the lower - bound

Given \(a = 12\) and \(b = 10\), the lower - bound is \(|12 - 10|=2\), so \(2\lt x\).

Step3: Calculate the upper - bound

The upper - bound is \(12 + 10 = 22\), so \(x\lt22\).

Answer:

A. \(2\lt x\lt22\)