To determine if each set of numbers can represent the sides of a triangle, we can apply the Triangle Inequality Theorem. This theorem states that for any three lengths to form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

Let's analyze each set of numbers:

1. 5, 4, and 1:

   • 5 + 4 > 1 (True)
   • 5 + 1 > 4 (True)
   • 4 + 1 > 5 (False)
   • Since one condition fails, No.

2. 3, 3, and 7:

   • 3 + 3 > 7 (False)
   • Since this fails, No.

3. 9, 7, and 14:

   • 9 + 7 > 14 (True)
   • 9 + 14 > 7 (True)
   • 7 + 14 > 9 (True)
   • All conditions are satisfied, Yes.

4. 4, 8, and 2:

   • 4 + 8 > 2 (True)
   • 4 + 2 > 8 (False)
   • Since one condition fails, No.

Thus, the answers for each set are:

• 5, 4, and 1: No
• 3, 3, and 7: No
• 9, 7, and 14: Yes
• 4, 8, and 2: No

Let me know if you'd like further explanations or have any questions.

Here are five related questions to explore further:

1. What is the Triangle Inequality Theorem, and why does it apply to triangle formation?
2. Can three sides of equal length always form a triangle?
3. What happens if two sides of a triangle are equal, but the third side is much longer?
4. How does the Triangle Inequality Theorem apply to real-world structures?
5. What are some examples of geometric shapes where the Triangle Inequality Theorem is essential?

Tip: To quickly check if lengths form a triangle, add each pair and compare with the remaining side. If all sums are greater, a triangle can be formed.