To determine if the side lengths of a triangle can be 3, 6, and 9, we can use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's check the three conditions based on the given lengths:
- \(3 + 6 > 9\) → \(9 > 9\) (not true)
- \(3 + 9 > 6\) → \(12 > 6\) (true)
- \(6 + 9 > 3\) → \(15 > 3\) (true)
Since the first condition does not hold true, the side lengths 3, 6, and 9 do not satisfy the triangle inequality theorem. Therefore, these side lengths cannot form a triangle.