To determine which set of three numbers could be the side lengths of a triangle, we use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. We can express this as:
- For sides [tex]\(a, b,\)[/tex] and [tex]\(c\)[/tex], with [tex]\(a \leq b \leq c\)[/tex],
1. [tex]\(a + b > c\)[/tex]
2. [tex]\(a + c > b\)[/tex]
3. [tex]\(b + c > a\)[/tex]
Given the options:
Option A: 3, 3, 8
- Sort the sides: 3, 3, 8
- Check the conditions:
- [tex]\(3 + 3 > 8\)[/tex] → [tex]\(6 > 8\)[/tex] (False)
- Because the first condition fails, 3, 3, and 8 cannot form a triangle.
Option B: 5, 5, 14
- Sort the sides: 5, 5, 14
- Check the conditions:
- [tex]\(5 + 5 > 14\)[/tex] → [tex]\(10 > 14\)[/tex] (False)
- Because the first condition fails, 5, 5, and 14 cannot form a triangle.
Option C: 3, 3, 2
- Sort the sides: 2, 3, 3
- Check the conditions:
- [tex]\(2 + 3 > 3\)[/tex] → [tex]\(5 > 3\)[/tex] (True)
- [tex]\(2 + 3 > 3\)[/tex] → [tex]\(5 > 3\)[/tex] (True)
- [tex]\(3 + 3 > 2\)[/tex] → [tex]\(6 > 2\)[/tex] (True)
- Since all conditions are satisfied, 3, 3, and 2 can indeed form a triangle.
Option D: 5, 5, 10
- Sort the sides: 5, 5, 10
- Check the conditions:
- [tex]\(5 + 5 > 10\)[/tex] → [tex]\(10 > 10\)[/tex] (False)
- Because the first condition fails, 5, 5, and 10 cannot form a triangle.
Among the given options, only set C: 3, 3, and 2 satisfies the triangle inequality theorem and hence can form a triangle. Therefore, the correct answer is:
C. 3, 3, 2
