Question

The Triangle inequality to see which set of side lengths will form a triangle

Answer 1

The Triangle Inequality Theorem states that for any triangle with side lengths \( a \), \( b \), and \( c \), the following conditions must hold:

1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

To determine whether a given set of side lengths can form a triangle, you simply need to check if these three conditions are satisfied for the lengths you have.

### Steps to Use the Triangle Inequality:

1. **Identify the side lengths**: Let's assume you have three lengths \( a \), \( b \), and \( c \).

2. **Check the Triangle Inequalities**:
- First check if \( a + b > c \).
- Then check if \( a + c > b \).
- Finally, check if \( b + c > a \).

3. **Conclusion**:
- If all three conditions are met, then the lengths can form a triangle.
- If any one of the conditions fails, then those lengths cannot form a triangle.

### Example:

Let's say you want to check if the lengths 5, 7, and 10 can form a triangle.

1. **Identify the sides**: \( a = 5 \), \( b = 7 \), \( c = 10 \).
2. **Check the inequalities**:
- \( 5 + 7 = 12 > 10 \) (True)
- \( 5 + 10 = 15 > 7 \) (True)
- \( 7 + 10 = 17 > 5 \) (True)
3. **Conclusion**: Since all conditions are satisfied, the lengths 5, 7, and 10 can form a triangle.

Feel free to provide specific side lengths to check, and I can help you verify if they can form a triangle!