Which one of the following cannot be represented by the three sides of a triangle?

Options

A. `5,9,11`
B. `3,7,11`
C. `7,10,13`
D. `3,8,9`

Explanation

To determine which set of three sides cannot form a triangle, we will use the triangle inequality theorem. According to this theorem, for any three sides \(a\), \(b\), and \(c\) to form a triangle, the following conditions must be satisfied:

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

Let's analyze each of the provided options step by step.

### Step 1: Analyze Option A: 5, 9, 11
- Check: \(5 + 9 > 11\) → \(14 > 11\) (True)
- Check: \(5 + 11 > 9\) → \(16 > 9\) (True)
- Check: \(9 + 11 > 5\) → \(20 > 5\) (True)

Since all conditions are satisfied, these sides can form a triangle.

### Step 2: Analyze Option B: 7, 10, 10
- Check: \(7 + 10 > 10\) → \(17 > 10\) (True)
- Check: \(7 + 10 > 10\) → \(17 > 10\) (True)
- Check: \(10 + 10 > 7\) → \(20 > 7\) (True)

Since all conditions are satisfied, these sides can form a triangle.

### Step 3: Analyze Option C: 8, 3, 9
- Check: \(8 + 3 > 9\) → \(11 > 9\) (True)
- Check: \(8 + 9 > 3\) → \(17 > 3\) (True)
- Check: \(3 + 9 > 8\) → \(12 > 8\) (True)

Since all conditions are satisfied, these sides can form a triangle.

### Step 4: Analyze Option D: 2, 3, 6
- Check: \(2 + 3 > 6\) → \(5 > 6\) (False)
- Check: \(2 + 6 > 3\) → \(8 > 3\) (True)
- Check: \(3 + 6 > 2\) → \(9 > 2\) (True)

Since the first condition is not satisfied, these sides cannot form a triangle.

### Conclusion
The set of sides that cannot form a triangle is **Option D: 2, 3, 6**.

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