Question

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

12, 10, 22
12, 10, 22

8, 1, 7
8, 1, 7

9, 4, 3
9, 4, 3

2, 7, 6

Answer 1

The Triangle Inequality Theorem states that for any three sides \( a \), \( b \), and \( c \) of a triangle, the following inequalities must hold true:

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

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

1. **12, 10, 22**
- \( 12 + 10 = 22 \) (not greater)
- \( 12 + 22 = 34 > 10 \)
- \( 10 + 22 = 32 > 12 \)
- **Does not form a triangle.**

2. **8, 1, 7**
- \( 8 + 1 = 9 > 7 \)
- \( 8 + 7 = 15 > 1 \)
- \( 1 + 7 = 8 \) (not greater)
- **Does not form a triangle.**

3. **9, 4, 3**
- \( 9 + 4 = 13 > 3 \)
- \( 9 + 3 = 12 > 4 \)
- \( 4 + 3 = 7 < 9 \) (not greater)
- **Does not form a triangle.**

4. **2, 7, 6**
- \( 2 + 7 = 9 > 6 \)
- \( 2 + 6 = 8 > 7 \)
- \( 7 + 6 = 13 > 2 \)
- **Forms a triangle.**

Based on the Triangle Inequality Theorem, the only group of side measures that will form a triangle is **2, 7, 6**.