Question

Determine which group aside measures will form a triangle
Two, seven, six
Nine, four, three
12, 10, 22
Eight, one, seven

Answer 1

To determine whether a group of three sides can form a triangle, we can use the triangle inequality theorem. This theorem states that for any three lengths \(a\), \(b\), and \(c\) to form a triangle, the following conditions must hold:

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

Let's analyze each group of measures:

1. **Two, seven, six**:
- \(2 + 7 = 9 > 6\) (True)
- \(2 + 6 = 8 > 7\) (True)
- \(7 + 6 = 13 > 2\) (True)
- Conclusion: This group can form a triangle.

2. **Nine, four, three**:
- \(9 + 4 = 13 > 3\) (True)
- \(9 + 3 = 12 > 4\) (True)
- \(4 + 3 = 7 < 9\) (False)
- Conclusion: This group cannot form a triangle.

3. **Twelve, ten, twenty-two**:
- \(12 + 10 = 22 \not > 22\) (False)
- \(12 + 22 = 34 > 10\) (True)
- \(10 + 22 = 32 > 12\) (True)
- Conclusion: This group cannot form a triangle.

4. **Eight, one, seven**:
- \(8 + 1 = 9 > 7\) (True)
- \(8 + 7 = 15 > 1\) (True)
- \(1 + 7 = 8 = 8\) (Not greater)
- Conclusion: This group cannot form a triangle.

### Summary:
- The group that can form a triangle: **Two, seven, six.**
- Groups that cannot form a triangle: **Nine, four, three; Twelve, ten, twenty-two; Eight, one, seven.**