To determine which set of numbers can represent the lengths of the sides of a triangle, we must use the triangle inequality theorem. The theorem states that for any three sides, [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] (with [tex]\(c\)[/tex] being the longest of the three), the following must be true:
[tex]\[ a + b > c \][/tex]
We will check each set of numbers with this theorem:
Option A: [tex]\(\{5, 5, 10\}\)[/tex]
Here, [tex]\(a = 5\)[/tex], [tex]\(b = 5\)[/tex], and [tex]\(c = 10\)[/tex]:
[tex]\[ 5 + 5 = 10 \][/tex]
Since [tex]\(a + b\)[/tex] is not greater than [tex]\(c\)[/tex], this set does not satisfy the triangle inequality theorem and cannot form a triangle.
Option B: [tex]\(\{2, 2, 6\}\)[/tex]
Here, [tex]\(a = 2\)[/tex], [tex]\(b = 2\)[/tex], and [tex]\(c = 6\)[/tex]:
[tex]\[ 2 + 2 = 4 \][/tex]
Since [tex]\(a + b\)[/tex] is less than [tex]\(c\)[/tex], this set does not satisfy the triangle inequality theorem and cannot form a triangle.
Option C: [tex]\(\{1, 8, 10\}\)[/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = 8\)[/tex], and [tex]\(c = 10\)[/tex]:
[tex]\[ 1 + 8 = 9 \][/tex]
Since [tex]\(a + b\)[/tex] is less than [tex]\(c\)[/tex], this set does not satisfy the triangle inequality theorem and cannot form a triangle.
Option D: [tex]\(\{5, 12, 13\}\)[/tex]
Here, [tex]\(a = 5\)[/tex], [tex]\(b = 12\)[/tex], and [tex]\(c = 13\)[/tex]:
[tex]\[ 5 + 12 = 17 \][/tex]
Since [tex]\(a + b\)[/tex] is greater than [tex]\(c\)[/tex], this set satisfies the triangle inequality theorem and can form a triangle.
Therefore, the set of numbers that may represent the lengths of the sides of a triangle is:
D. [tex]\(\{5, 12, 13\}\)[/tex]
