To create a set of five side lengths, where no matter which three you choose, you cannot form a triangle, we must apply the triangle inequality theorem. This theorem states that, for any three sides to form a triangle, the sum of the lengths of any two sides must always be greater than the length of the remaining side.

Thus, we need to select five side lengths such that for any combination of three sides, the sum of the two smaller sides is less than or equal to the third (largest) side. This ensures no triangle can be formed.

One example of such a set is:

$\{1, 2, 4, 8, 16\}$

Verifying:

For any combination of three sides, the sum of the two smaller sides will always be less than or equal to the largest side:

• $1 + 2 = 3$, which is less than $4$,
• $1 + 4 = 5$, which is less than $8$,
• $2 + 4 = 6$, which is less than $8$,
• $1 + 8 = 9$, which is less than $16$, and so on for all other combinations.

Thus, no combination of three sides from this set can form a triangle.

Would you like further explanation on this, or have any questions?

Related Questions:

1. What is the triangle inequality theorem in more detail?
2. Can you explain other examples of sets that do not form triangles?
3. How can you check if a set of three numbers forms a triangle?
4. What would happen if one side length was slightly larger? Would a triangle be possible?
5. What is the difference between a triangle's inequality and equality conditions?

Tip:

Always check all combinations of three sides when applying the triangle inequality theorem to ensure no triangles can form.