To determine which set of numbers can represent the lengths of the sides of a triangle, we need to use the triangle inequality theorem. This theorem states that for three sides to form a triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
Let's analyze each set of numbers step-by-step:
### Option A: [tex]\(\{1, 8, 10\}\)[/tex]
1. [tex]\(1 + 8 > 10\)[/tex]: [tex]\(9 > 10\)[/tex] (False)
2. [tex]\(1 + 10 > 8\)[/tex]: [tex]\(11 > 8\)[/tex] (True)
3. [tex]\(8 + 10 > 1\)[/tex]: [tex]\(18 > 1\)[/tex] (True)
Since not all conditions are satisfied (specifically, [tex]\(1 + 8 > 10\)[/tex] is false), this set of numbers cannot form a triangle.
### Option B: [tex]\(\{5, 5, 10\}\)[/tex]
1. [tex]\(5 + 5 > 10\)[/tex]: [tex]\(10 > 10\)[/tex] (False)
2. [tex]\(5 + 10 > 5\)[/tex]: [tex]\(15 > 5\)[/tex] (True)
3. [tex]\(5 + 10 > 5\)[/tex]: [tex]\(15 > 5\)[/tex] (True)
Since not all conditions are satisfied (specifically, [tex]\(5 + 5 > 10\)[/tex] is false), this set of numbers cannot form a triangle.
### Option C: [tex]\(\{5, 12, 13\}\)[/tex]
1. [tex]\(5 + 12 > 13\)[/tex]: [tex]\(17 > 13\)[/tex] (True)
2. [tex]\(5 + 13 > 12\)[/tex]: [tex]\(18 > 12\)[/tex] (True)
3. [tex]\(12 + 13 > 5\)[/tex]: [tex]\(25 > 5\)[/tex] (True)
Since all conditions are satisfied, this set of numbers can form a triangle.
### Option D: [tex]\(\{2, 2, 6\}\)[/tex]
1. [tex]\(2 + 2 > 6\)[/tex]: [tex]\(4 > 6\)[/tex] (False)
2. [tex]\(2 + 6 > 2\)[/tex]: [tex]\(8 > 2\)[/tex] (True)
3. [tex]\(2 + 6 > 2\)[/tex]: [tex]\(8 > 2\)[/tex] (True)
Since not all conditions are satisfied (specifically, [tex]\(2 + 2 > 6\)[/tex] is false), this set of numbers cannot form a triangle.
Based on the analysis, the only set of numbers that can represent the lengths of the sides of a triangle is:
C. [tex]\(\{5, 12, 13\}\)[/tex]
