Question

Alisha wants to build a miniature bridge that will include a triangle made of wooden pieces. Which lengths of wood can she use to form a triangle? All lengths are given in inches.(1 point)
Responses

12, 4, and 16
12, 4, and 16

12, 16, and 28
12, 16, and 28

3, 16, and 12
3, 16, and 12

27, 12, and 16

Answer 1

To determine which sets of lengths can form a triangle, we can use the triangle inequality theorem. The triangle inequality states that for any three sides \( a \), \( b \), and \( c \), the following must hold:

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

Let's evaluate each set of lengths:

1. **12, 4, and 16**:
- \( 12 + 4 = 16 \) (not greater)
- \( 12 + 16 = 28 \) (greater)
- \( 4 + 16 = 20 \) (greater)
- This set does not satisfy the triangle inequality.

2. **12, 16, and 28**:
- \( 12 + 16 = 28 \) (not greater)
- \( 12 + 28 = 40 \) (greater)
- \( 16 + 28 = 44 \) (greater)
- This set does not satisfy the triangle inequality.

3. **3, 16, and 12**:
- \( 3 + 12 = 15 \) (not greater than 16)
- \( 3 + 16 = 19 \) (greater)
- \( 12 + 16 = 28 \) (greater)
- This set does not satisfy the triangle inequality.

4. **27, 12, and 16**:
- \( 27 + 12 = 39 \) (greater)
- \( 27 + 16 = 43 \) (greater)
- \( 12 + 16 = 28 \) (not greater than 27)
- This set does not satisfy the triangle inequality.

None of the provided sets of lengths can form a triangle based on the triangle inequality theorem.