Question

Can the sides of a triangle have lengths 1, 3, and 11?

Answer 1

To determine whether a set of lengths 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:

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

In this case, we have side lengths \( 1 \), \( 3 \), and \( 11 \). Let's assign these values to \( a \), \( b \), and \( c \):

- Let \( a = 1 \)
- Let \( b = 3 \)
- Let \( c = 11 \)

Now we will check the inequalities:

1. \( a + b > c \):
\[
1 + 3 > 11 \quad \Rightarrow \quad 4 > 11 \quad \text{(false)}
\]

2. \( a + c > b \):
\[
1 + 11 > 3 \quad \Rightarrow \quad 12 > 3 \quad \text{(true)}
\]

3. \( b + c > a \):
\[
3 + 11 > 1 \quad \Rightarrow \quad 14 > 1 \quad \text{(true)}
\]

Since the first inequality \( a + b > c \) is false, the lengths \( 1 \), \( 3 \), and \( 11 \) do not satisfy the triangle inequality. Therefore, these lengths cannot form a triangle.