Question

Which set of side lengths cannot form a triangle?
(4 Points)

3 cm, 4 cm, 6 cm

5 cm, 7 cm, 12 cm

8 cm, 10 cm, 18 cm

2 cm, 4 cm, 7 cm

Answer 1

To determine which set of side lengths cannot form a triangle, we can apply the triangle inequality theorem, which states that for any triangle with side lengths \(a\), \(b\), and \(c\), the following conditions must hold:

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

Let's evaluate each set of side lengths:

1. **3 cm, 4 cm, 6 cm**:
- \(3 + 4 = 7 > 6\)
- \(3 + 6 = 9 > 4\)
- \(4 + 6 = 10 > 3\)
- This set can form a triangle.

2. **5 cm, 7 cm, 12 cm**:
- \(5 + 7 = 12\), which is not greater than 12.
- This set cannot form a triangle.

3. **8 cm, 10 cm, 18 cm**:
- \(8 + 10 = 18\), which is not greater than 18.
- This set cannot form a triangle.

4. **2 cm, 4 cm, 7 cm**:
- \(2 + 4 = 6 < 7\), which is not greater than 7.
- This set cannot form a triangle.

Based on the results, the sets that cannot form a triangle are:

- **5 cm, 7 cm, 12 cm**
- **8 cm, 10 cm, 18 cm**
- **2 cm, 4 cm, 7 cm**

However, since the question asks for **which set** of side lengths cannot form a triangle, the set with the smallest sum violating the triangle inequality is:

**5 cm, 7 cm, 12 cm** is the simplest answer as it directly fails the inequality test of forming a triangle.

Therefore, the answer is **5 cm, 7 cm, 12 cm**.