Question

Can the sides of a triangle have lengths 2, 16, and 17?

Answer 1

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:

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

Let's denote the sides as follows:
- \(a = 2\)
- \(b = 16\)
- \(c = 17\)

Now, we will check each of the triangle inequalities:

1. \(a + b > c\):
\[
2 + 16 > 17 \quad \text{(which simplifies to)} \quad 18 > 17 \quad \text{(True)}
\]

2. \(a + c > b\):
\[
2 + 17 > 16 \quad \text{(which simplifies to)} \quad 19 > 16 \quad \text{(True)}
\]

3. \(b + c > a\):
\[
16 + 17 > 2 \quad \text{(which simplifies to)} \quad 33 > 2 \quad \text{(True)}
\]

Since all three conditions of the triangle inequality are satisfied, the lengths 2, 16, and 17 can indeed form a triangle.