Question

Can the sides of a triangle have lengths 5, 1, and 4?

Answer 1

To determine whether three lengths can form a triangle, we can use the triangle inequality theorem. This theorem states that for any triangle with side lengths \(a\), \(b\), and \(c\), the following conditions must be satisfied:

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

Given the lengths 5, 1, and 4, we can let \(a = 5\), \(b = 1\), and \(c = 4\). Now, we will check each of the triangle inequality conditions:

1. \(a + b > c\)
\[
5 + 1 > 4 \quad \text{(6 > 4, which is true)}
\]

2. \(a + c > b\)
\[
5 + 4 > 1 \quad \text{(9 > 1, which is true)}
\]

3. \(b + c > a\)
\[
1 + 4 > 5 \quad \text{(5 > 5, which is false)}
\]

Since the third condition (\(b + c > a\)) is not satisfied, the lengths 5, 1, and 4 cannot form a triangle. Thus, the answer is **no**.