Answer :

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To determine whether a given set of three numbers can be the side lengths of a triangle, we use the triangle inequality theorem.

The triangle inequality theorem states that for any three sides of a potential triangle [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:

1. [tex]\(a + b > c\)[/tex]
2. [tex]\(b + c > a\)[/tex]
3. [tex]\(c + a > b\)[/tex]

We'll apply these rules to each of the given sets:

Set B: 4, 10, 3

1. [tex]\(4 + 10 > 3\)[/tex]
- [tex]\(14 > 3\)[/tex] (True)
2. [tex]\(10 + 3 > 4\)[/tex]
- [tex]\(13 > 4\)[/tex] (True)
3. [tex]\(3 + 4 > 10\)[/tex]
- [tex]\(7 > 10\)[/tex] (False)

Since the third condition is false, the set (4, 10, 3) cannot form a triangle.

Set C: 4, 7, 11

1. [tex]\(4 + 7 > 11\)[/tex]
- [tex]\(11 > 11\)[/tex] (False)
2. [tex]\(7 + 11 > 4\)[/tex]
- [tex]\(18 > 4\)[/tex] (True)
3. [tex]\(11 + 4 > 7\)[/tex]
- [tex]\(15 > 7\)[/tex] (True)

Since the first condition is false, the set (4, 7, 11) cannot form a triangle.

Set P: 4, 10, 6

1. [tex]\(4 + 10 > 6\)[/tex]
- [tex]\(14 > 6\)[/tex] (True)
2. [tex]\(10 + 6 > 4\)[/tex]
- [tex]\(16 > 4\)[/tex] (True)
3. [tex]\(6 + 4 > 10\)[/tex]
- [tex]\(10 > 10\)[/tex] (False)

Since the third condition is false, the set (4, 10, 6) cannot form a triangle.

Based on these evaluations:

- Set B: 4, 10, 3 → False (Not a triangle)
- Set C: 4, 7, 11 → False (Not a triangle)
- Set P: 4, 10, 6 → False (Not a triangle)

None of these sets of numbers can be the side lengths of a triangle.

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