Question

two sides of a triangle have the measures 12 and 10. find the range of possible measures for the third side.

a. 2 < x < 22
b. 10 < x < 12
c. 2 < x < 12
d. 10 < x < 22

Explanation:

Step1: Recall triangle - inequality theorem

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Let the sides of the triangle be \(a = 12\), \(b = 10\), and \(c=x\). Then \(a + b>c\), \(a + c>b\), and \(b + c>a\). Also, \(|a - b|

Step2: Calculate the lower - bound

The difference between the two given sides gives the lower - bound of the third side. \(|12 - 10|=2\), so \(x>2\).

Step3: Calculate the upper - bound

The sum of the two given sides gives the upper - bound of the third side. \(12 + 10 = 22\), so \(x<22\).

Answer:

A. \(2