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:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
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:
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\( a + b > c \): \[ 1 + 3 > 11 \quad \Rightarrow \quad 4 > 11 \quad \text{(false)} \]
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\( a + c > b \): \[ 1 + 11 > 3 \quad \Rightarrow \quad 12 > 3 \quad \text{(true)} \]
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\( b + c > a \): \[ 3 + 11 > 1 \quad \Rightarrow \quad 14 > 1 \quad \text{(true)} \]
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.