Question

Which sets of side lengths form a triangle? All lengths are given in inches.

(Choose 3 answers.) 

(1 point)
Responses

9, 17, 11
9, 17, 11

11, 19, 9
11, 19, 9

18, 10, 8
18, 10, 8

18, 11, 8

Answer 1

To determine whether a set of side lengths can form a triangle, we can use the triangle inequality theorem, which states that for any triangle with sides of lengths \( a \), \( b \), and \( c \):

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

We will apply this theorem to each set of lengths provided.

1. **For the set (9, 17, 11)**:
- \( 9 + 11 > 17 \) → \( 20 > 17 \) (True)
- \( 9 + 17 > 11 \) → \( 26 > 11 \) (True)
- \( 11 + 17 > 9 \) → \( 28 > 9 \) (True)
- **This set can form a triangle.**

2. **For the set (11, 19, 9)**:
- \( 11 + 9 > 19 \) → \( 20 > 19 \) (True)
- \( 11 + 19 > 9 \) → \( 30 > 9 \) (True)
- \( 19 + 9 > 11 \) → \( 28 > 11 \) (True)
- **This set can form a triangle.**

3. **For the set (18, 10, 8)**:
- \( 18 + 10 > 8 \) → \( 28 > 8 \) (True)
- \( 18 + 8 > 10 \) → \( 26 > 10 \) (True)
- \( 10 + 8 > 18 \) → \( 18 > 18 \) (False)
- **This set cannot form a triangle.**

4. **For the set (18, 11, 8)**:
- \( 18 + 11 > 8 \) → \( 29 > 8 \) (True)
- \( 18 + 8 > 11 \) → \( 26 > 11 \) (True)
- \( 11 + 8 > 18 \) → \( 19 > 18 \) (True)
- **This set can form a triangle.**

Based on the analysis, the sets of side lengths that can form a triangle are:

1. **(9, 17, 11)**
2. **(11, 19, 9)**
3. **(18, 11, 8)**

So the correct responses are **(9, 17, 11)**, **(11, 19, 9)**, and **(18, 11, 8)**.