To determine which set of numbers does NOT represent the lengths of the sides of a triangle, we need to use the triangle inequality theorem. This theorem states that for any three sides of a triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. We will check each set of numbers:
### Set A: [tex]\(\{7, 5, 6\}\)[/tex]
1. [tex]\(7 + 5 = 12\)[/tex], which is greater than [tex]\(6\)[/tex].
2. [tex]\(7 + 6 = 13\)[/tex], which is greater than [tex]\(5\)[/tex].
3. [tex]\(5 + 6 = 11\)[/tex], which is greater than [tex]\(7\)[/tex].
Since all three conditions are met, the set [tex]\(\{7, 5, 6\}\)[/tex] can form a triangle.
### Set B: [tex]\(\{6, 8, 11\}\)[/tex]
1. [tex]\(6 + 8 = 14\)[/tex], which is greater than [tex]\(11\)[/tex].
2. [tex]\(6 + 11 = 17\)[/tex], which is greater than [tex]\(8\)[/tex].
3. [tex]\(8 + 11 = 19\)[/tex], which is greater than [tex]\(6\)[/tex].
Since all three conditions are met, the set [tex]\(\{6, 8, 11\}\)[/tex] can form a triangle.
### Set C: [tex]\(\{7, 18, 11\}\)[/tex]
1. [tex]\(7 + 18 = 25\)[/tex], which is greater than [tex]\(11\)[/tex].
2. [tex]\(7 + 11 = 18\)[/tex], which is equal to [tex]\(18\)[/tex].
3. [tex]\(18 + 11 = 29\)[/tex], which is greater than [tex]\(7\)[/tex].
Because [tex]\(7 + 11\)[/tex] is equal to [tex]\(18\)[/tex], not greater than it, this set does not satisfy the triangle inequality theorem.
### Set D: [tex]\(\{9, 12, 19\}\)[/tex]
1. [tex]\(9 + 12 = 21\)[/tex], which is greater than [tex]\(19\)[/tex].
2. [tex]\(9 + 19 = 28\)[/tex], which is greater than [tex]\(12\)[/tex].
3. [tex]\(12 + 19 = 31\)[/tex], which is greater than [tex]\(9\)[/tex].
Since all three conditions are met, the set [tex]\(\{9, 12, 19\}\)[/tex] can form a triangle.
### Conclusion
The set [tex]\(\{7, 18, 11\}\)[/tex] does not satisfy the triangle inequality theorem, as the sum of some sides does not exceed the third side. Therefore, the set of numbers that does NOT represent the lengths of the sides of a triangle is:
C. [tex]\(\{7, 18, 11\}\)[/tex]
