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
x>7
7<x<17
given a triangle with side lengths of 5 feet and 12 feet, the possible whole - number measures of the third side (8 feet).

1. two sides of a triangle have the following sides, find the range for the measures of the third side.

a. 14, 11, 3<x<25
a + b = 14+11 =
25 a - b =
14 - 11 =
3<x<25
b. 6, 10, 4<x<16
a - b+ a + 10
10 - 6+ a - 6
= 4<x<16
d. 9, 5,
a + b = a + 5
9 + 5 a + b =
e. 5, 8,
f. 6,
h. 11, 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 length be \(x\). Then \(|a - b|\lt x\lt a + b\).

Step2: Solve for part d

Given \(a = 9\) and \(b = 5\). First, calculate \(a + b=9 + 5=14\) and \(|a - b|=|9 - 5| = 4\). So, \(4\lt x\lt14\).

Step3: Solve for part e

Given \(a = 5\) and \(b = 8\). Calculate \(a + b=5 + 8 = 13\) and \(|a - b|=|8 - 5|=3\). So, \(3\lt x\lt13\).

Step4: Solve for part h

Given \(a = 11\) and \(b = 8\). Calculate \(a + b=11+8 = 19\) and \(|a - b|=|11 - 8| = 3\). So, \(3\lt x\lt19\).

Answer:

d. \(4\lt x\lt14\)
e. \(3\lt x\lt13\)
h. \(3\lt x\lt19\)