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 sum of the lengths of any two sides of a triangle must be greater than the length of the third side, and the difference between the lengths of any two sides must be less than the length of the third side.
Let the two given sides be \(a = 12\) and \(b = 10\).

Step2: Calculate the upper - bound

The sum of the two given sides gives the upper - bound of the third side. So \(x\lt a + b\), substituting \(a = 12\) and \(b = 10\), we get \(x\lt12 + 10=22\).

Step3: Calculate the lower - bound

The difference of the two given sides gives the lower - bound of the third side. So \(|a - b|\lt x\), \(|12 - 10|=2\), and \(2\lt x\).

Answer:

A. \(2 < x < 22\)