Question

I triangle has size 12.5 inches and 22 inches what side length below can create a triangle with these two sides
4 inches
9 inches
11 inches
35 inches

Answer 1

To determine which of the given side lengths can form a triangle with the two sides of lengths 12.5 inches and 22 inches, we can use the triangle inequality theorem. This theorem states that for three sides \(a\), \(b\), and \(c\) to form a triangle, the following conditions must be met:

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

In our case, we have two sides \(a = 12.5\) inches and \(b = 22\) inches, and we need to test each possible third side (let's call it \(c\)) from the provided options: 4 inches, 9 inches, 11 inches, and 35 inches.

### Testing Each Side Length

1. **For \(c = 4\) inches**:
- \(12.5 + 4 > 22\) → \(16.5 > 22\) (False)
- \(12.5 + 22 > 4\) → \(34.5 > 4\) (True)
- \(22 + 4 > 12.5\) → \(26 > 12.5\) (True)
- **Conclusion**: Fails the first condition. **Cannot form a triangle**.

2. **For \(c = 9\) inches**:
- \(12.5 + 9 > 22\) → \(21.5 > 22\) (False)
- \(12.5 + 22 > 9\) → \(34.5 > 9\) (True)
- \(22 + 9 > 12.5\) → \(31 > 12.5\) (True)
- **Conclusion**: Fails the first condition. **Cannot form a triangle**.

3. **For \(c = 11\) inches**:
- \(12.5 + 11 > 22\) → \(23.5 > 22\) (True)
- \(12.5 + 22 > 11\) → \(34.5 > 11\) (True)
- \(22 + 11 > 12.5\) → \(33 > 12.5\) (True)
- **Conclusion**: All conditions are satisfied. **Can form a triangle**.

4. **For \(c = 35\) inches**:
- \(12.5 + 35 > 22\) → \(47.5 > 22\) (True)
- \(12.5 + 22 > 35\) → \(34.5 > 35\) (False)
- \(22 + 35 > 12.5\) → \(57 > 12.5\) (True)
- **Conclusion**: Fails the second condition. **Cannot form a triangle**.

### Final Result
The only side length among the options that can create a triangle with the other two sides (12.5 inches and 22 inches) is **11 inches**.