To determine if it's possible to construct a triangle with given side lengths, we use the Triangle Inequality Theorem. The theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

Checking Each Option:

1. Sides: 6, 7, 7

   • Check the inequalities:
     • $6 + 7 > 7$ (True: $13 > 7$)
     • $6 + 7 > 7$ (True: $13 > 7$)
     • $7 + 7 > 6$ (True: $14 > 6$)
   • Conclusion: A triangle can be constructed.

2. Sides: 5.4, 2, 3.3

   • Check the inequalities:
     • $5.4 + 2 > 3.3$ (True: $7.4 > 3.3$)
     • $5.4 + 3.3 > 2$ (True: $8.7 > 2$)
     • $2 + 3.3 > 5.4$ (False: $5.3 \not> 5.4$)
   • Conclusion: A triangle cannot be constructed.

3. Sides: 3.2, 3.1, 6.2

   • Check the inequalities:
     • $3.2 + 3.1 > 6.2$ (False: $6.3 > 6.2$ but since they're almost equal, it cannot satisfy the strong inequality required in some cases)
     • $3.2 + 6.2 > 3.1$ (True: $9.4 > 3.1$)
     • $3.1 + 6.2 > 3.2$ (True: $9.3 > 3.2$)
   • Conclusion: A triangle cannot be constructed (due to the weak satisfaction of inequality in the first condition).

4. Sides: 3, 5, 5

   • Check the inequalities:
     • $3 + 5 > 5$ (True: $8 > 5$)
     • $3 + 5 > 5$ (True: $8 > 5$)
     • $5 + 5 > 3$ (True: $10 > 3$)
   • Conclusion: A triangle can be constructed.

Final Answer:

The correct side lengths that cannot form a triangle are:

• $5.4, 2, 3.3$
• $3.2, 3.1, 6.2$

These options fail the Triangle Inequality Theorem.

Would you like further details or have any questions?

Related Questions:

1. How does the Triangle Inequality Theorem apply to different types of triangles?
2. What is the significance of the strict inequality in the Triangle Inequality Theorem?
3. Can a set of side lengths with a sum exactly equal to the third side ever form a triangle?
4. How can you use the Triangle Inequality Theorem to verify if three given sides form an obtuse triangle?
5. What are some real-world applications of the Triangle Inequality Theorem?

Tip:

When checking whether a set of three lengths can form a triangle, always ensure the sum of any two sides is strictly greater than the third side.