Question

Apply the Triangle Inequality Theorem and determine which group of side measures will form a triangle.(1 point)
Responses

9, 4, 3
9, 4, 3

2, 7, 6
2, 7, 6

12, 10, 22
12, 10, 22

8, 1, 7
8, 1, 7

Answer 1

To determine if a group of side measures can form a triangle, we can apply the Triangle Inequality Theorem. This theorem states that for any triangle with sides of lengths a, b, and c, the following conditions must all be true:

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

Let's apply this to each group of side measures:

1. **9, 4, 3**
- \( 9 + 4 = 13 > 3 \) (True)
- \( 9 + 3 = 12 > 4 \) (True)
- \( 4 + 3 = 7 > 9 \) (False)
- **Conclusion**: Cannot form a triangle.

2. **2, 7, 6**
- \( 2 + 7 = 9 > 6 \) (True)
- \( 2 + 6 = 8 > 7 \) (True)
- \( 7 + 6 = 13 > 2 \) (True)
- **Conclusion**: Can form a triangle.

3. **12, 10, 22**
- \( 12 + 10 = 22 > 22 \) (False; must be greater, not equal)
- **Conclusion**: Cannot form a triangle.

4. **8, 1, 7**
- \( 8 + 1 = 9 > 7 \) (True)
- \( 8 + 7 = 15 > 1 \) (True)
- \( 1 + 7 = 8 > 8 \) (False; must be greater, not equal)
- **Conclusion**: Cannot form a triangle.

Based on the Triangle Inequality Theorem, only the group **2, 7, 6** can form a triangle.