To determine if the lengths 2, 16, and 17 can form a triangle, we must use the triangle inequality theorem. The theorem states that for any triangle with sides of lengths \(a\), \(b\), and \(c\), the following conditions must be satisfied:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
Let's denote the sides as follows:
- \(a = 2\)
- \(b = 16\)
- \(c = 17\)
Now, we will check each of the triangle inequalities:
-
\(a + b > c\): \[ 2 + 16 > 17 \quad \text{(which simplifies to)} \quad 18 > 17 \quad \text{(True)} \]
-
\(a + c > b\): \[ 2 + 17 > 16 \quad \text{(which simplifies to)} \quad 19 > 16 \quad \text{(True)} \]
-
\(b + c > a\): \[ 16 + 17 > 2 \quad \text{(which simplifies to)} \quad 33 > 2 \quad \text{(True)} \]
Since all three conditions of the triangle inequality are satisfied, the lengths 2, 16, and 17 can indeed form a triangle.