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)%0D%0AResponses%0D%0A%0D%0A3, 16, and 12%0D%0A3, 16, and 12%0D%0A%0D%0A12, 16, and 28%0D%0A12, 16, and 28%0D%0A%0D%0A27, 12, and 16%0D%0A27, 12, and 16%0D%0A%0D%0A12, 4, and 16

Answer 1

To determine which sets of lengths can form a triangle, we can use the triangle inequality theorem, which states that for any three lengths \( a \), \( b \), and \( c \) (where \( a \leq b \leq c \)), the following must be true:

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

This simplifies to only needing to check the condition \( a + b > c \) when the lengths are ordered from smallest to largest.

Let's evaluate each set of lengths:

1. **3, 16, and 12**:
- Ordered: \( 3, 12, 16 \)
- Checking: \( 3 + 12 = 15 \) (not greater than 16)
- This set **cannot** form a triangle.

2. **12, 16, and 28**:
- Ordered: \( 12, 16, 28 \)
- Checking: \( 12 + 16 = 28 \) (not greater than 28)
- This set **cannot** form a triangle.

3. **27, 12, and 16**:
- Ordered: \( 12, 16, 27 \)
- Checking: \( 12 + 16 = 28 \) (greater than 27)
- This set **can** form a triangle.

4. **12, 4, and 16**:
- Ordered: \( 4, 12, 16 \)
- Checking: \( 4 + 12 = 16 \) (not greater than 16)
- This set **cannot** form a triangle.

The only set of lengths that can form a triangle is **27, 12, and 16**.