Question

learning goal: students will be able to apply the triangle inequality theorem to determine the range of possible lengths for the third side of a triangle when given the other two side lengths. key terms: angle - side relationship, inequality, triangle, segment, triangle inequality theorem. example: if the measures of two sides of a triangle are 5 feet and 12 feet, what is the least possible whole - number measure for the third side? the triangle inequality: the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. 5 + 12>x, 17>x, x<17; x + 5>12, - 5 - 5, x>7. whole - numbers greater than 7 and less than 17. given a triangle with two side lengths of 5 feet and 12 feet, the least possible whole - number measure for the third side is 8 feet. note: 7 and 17 are not included. two sides of a triangle have the following sides, find the range for the possible measures of: a. 14, 11, __; b. 6, 10, ; c. 17, 19, ; e. 5, 8, __

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 two given side - lengths be \(a\) and \(b\), and the third side be \(x\). Then \(|a - b|\lt x\lt a + b\).

Step2: Solve for \(a = 14\) and \(b = 11\)

First, calculate \(a + b=14 + 11=25\) and \(|a - b|=|14 - 11| = 3\). So the range for the third side \(x\) is \(3\lt x\lt25\).

Step3: Solve for \(a = 6\) and \(b = 10\)

Calculate \(a + b=6+10 = 16\) and \(|a - b|=|6 - 10|=4\). So the range for the third side \(x\) is \(4\lt x\lt16\).

Step4: Solve for \(a = 5\) and \(b = 8\)

Calculate \(a + b=5 + 8=13\) and \(|a - b|=|5 - 8| = 3\). So the range for the third side \(x\) is \(3\lt x\lt13\).

Answer:

a. \(3\lt x\lt25\)
b. \(4\lt x\lt16\)
e. \(3\lt x\lt13\)