Understanding the Conditions for Forming a Triangle

To determine if a set of lengths can form a triangle, we must adhere to the triangle inequality theorem. This fundamental principle holds that the sum of the lengths of any two sides of a triangle must be greater than the third side. This ensures that the given lengths can indeed form a triangle, meaning it is possible to construct a triangle with those side lengths.

Analyzing Given Sets of Lengths

Let's analyze the given sets of lengths to see if they can form triangles:

Sides: 3, 5, 10 Analysis: For these lengths to form a triangle, the sum of the two shorter sides (3 and 5) must be greater than the longest side (10). However, (3 5 8), which is less than 10. Therefore, the lengths 3, 5, and 10 cannot form a triangle. Conclusion: No triangle. Sides: 13, 5, 11 Analysis: We need to check all combinations of side lengths to ensure the conditions of the triangle inequality theorem are met. The longest side here is 13. Checking the sum of the other two sides: 13 5 11 (13 16, which is false) 11 13 5 (11 18, which is false) 5 13 11 (5 24, which is false) Conclusion: No triangle. Sides: 9, 12, 24 Analysis: Here, the longest side is 24. Checking the sums of the other pairs: 9 12 21, which is less than 24 Conclusion: No triangle. Sides: 5, 2, 8 Analysis: The longest side is 8. Checking the sums of the other pairs: 5 2 7, which is less than 8 Conclusion: No triangle.

Based on our analysis, none of the provided sets of lengths can form a triangle, except for the set 13, 5, 11, which does adhere to the triangle inequality theorem.

Key Takeaways and Additional Insights

1. **Triangle Inequality Theorem**: The sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side. This is a crucial condition for determining if a set of lengths can form a triangle.

2. **Checking All Combinations**: When verifying if a set of lengths can form a triangle, it's essential to check all possible combinations of the lengths to ensure the triangle inequality theorem is met.

3. **Practical Application**: Understanding these conditions can be useful in various fields, from geometry and mathematics to engineering and construction, ensuring that certain measurements and configurations are feasible.

By adhering to these principles, you can systematically determine the set of possible lengths of the sides of a triangle and apply this knowledge in real-world scenarios.