Question

Which set of side length does not form a triangle?
A. 18, 11, 8
B. 11, 9, 9
C. 9, 17, 11
D. 18, 10, 8

Answer 1

To determine which set of side lengths does not form a triangle, we can use the triangle inequality theorem. According to this theorem, for any three sides \(a\), \(b\), and \(c\) of a triangle, the following conditions must hold:

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

We will check each option:

**A. 18, 11, 8**
- \(18 + 11 > 8 \quad \text{(True)}\)
- \(18 + 8 > 11 \quad \text{(True)}\)
- \(11 + 8 > 18 \quad \text{(False)}\)

Since one of the inequalities is false, these side lengths cannot form a triangle.

**B. 11, 9, 9**
- \(11 + 9 > 9 \quad \text{(True)}\)
- \(11 + 9 > 9 \quad \text{(True)}\)
- \(9 + 9 > 11 \quad \text{(True)}\)

All inequalities hold, so these side lengths can form a triangle.

**C. 9, 17, 11**
- \(9 + 17 > 11 \quad \text{(True)}\)
- \(9 + 11 > 17 \quad \text{(False)}\)
- \(17 + 11 > 9 \quad \text{(True)}\)

One inequality fails, so these side lengths cannot form a triangle.

**D. 18, 10, 8**
- \(18 + 10 > 8 \quad \text{(True)}\)
- \(18 + 8 > 10 \quad \text{(True)}\)
- \(10 + 8 > 18 \quad \text{(False)}\)

One inequality fails, so these side lengths cannot form a triangle.

From our analysis:

- **Option A** (18, 11, 8) cannot form a triangle.
- **Option C** (9, 17, 11) cannot form a triangle.
- **Option D** (18, 10, 8) cannot form a triangle.

Since all options except B result in at least one false inequality, the options that cannot form a triangle are **A, C, and D**. Since the question asks for a single set, **A. 18, 11, 8** is the first mentioned set that does not form a triangle.