Question

Apply the Triangle Inequality Theorem and determine which group of side measures will form a triangle. Responses: 9, 4, 3 9, 4, 3 2, 7, 6 2, 7, 6 12, 10, 22 12, 10, 22 8, 1, 7

Answer 1

To determine whether a set of side measures can form a triangle, we can use the Triangle Inequality Theorem. According to the theorem, for any three side lengths \( a, b, c \), the following must hold true:

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

Let's check each of the groups of side measures one by one:

1. **For the group \( 9, 4, 3 \)**:
- \( 9 + 4 = 13 > 3 \) (True)
- \( 9 + 3 = 12 > 4 \) (True)
- \( 4 + 3 = 7 > 9 \) (False)
- **This group does NOT form a triangle.**

2. **For the group \( 2, 7, 6 \)**:
- \( 2 + 7 = 9 > 6 \) (True)
- \( 2 + 6 = 8 > 7 \) (True)
- \( 7 + 6 = 13 > 2 \) (True)
- **This group DOES form a triangle.**

3. **For the group \( 12, 10, 22 \)**:
- \( 12 + 10 = 22 \not> 22 \) (False)
- **This group does NOT form a triangle.**

4. **For the group \( 8, 1, 7 \)**:
- \( 8 + 1 = 9 > 7 \) (True)
- \( 8 + 7 = 15 > 1 \) (True)
- \( 1 + 7 = 8 \not> 8 \) (False)
- **This group does NOT form a triangle.**

### Summary:
- \( 9, 4, 3 \): **Does not form a triangle**
- \( 2, 7, 6 \): **Forms a triangle**
- \( 12, 10, 22 \): **Does not form a triangle**
- \( 8, 1, 7 \): **Does not form a triangle**