Question

Can the sides of a triangle have lengths 6, 14, 18. Remember to use the Triangle Inequality Theorem.

Answer 1

To determine if the lengths 6, 14, and 18 can form a triangle, we need to apply the Triangle Inequality Theorem. This theorem states that for any triangle with sides of lengths \(a\), \(b\), and \(c\), the following inequalities must hold true:

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

Let's assign the side lengths:
- \(a = 6\)
- \(b = 14\)
- \(c = 18\)

Now we will check each of the inequalities:

1. Check \(a + b > c\):
\[
6 + 14 > 18 \implies 20 > 18 \quad \text{(True)}
\]

2. Check \(a + c > b\):
\[
6 + 18 > 14 \implies 24 > 14 \quad \text{(True)}
\]

3. Check \(b + c > a\):
\[
14 + 18 > 6 \implies 32 > 6 \quad \text{(True)}
\]

Since all three inequalities are satisfied, the lengths 6, 14, and 18 can indeed form a triangle.