To determine if a triangle can be formed with sides of lengths 3, 18, and 18, we need to apply the triangle inequality theorem. This theorem states that for any triangle with sides of lengths \( a \), \( b \), and \( c \):
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Let's assign the lengths to \( a = 3 \), \( b = 18 \), and \( c = 18 \), and then check each condition:
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\( a + b > c \): \[ 3 + 18 > 18 \] \[ 21 > 18 \quad \text{(True)} \]
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\( a + c > b \): \[ 3 + 18 > 18 \] \[ 21 > 18 \quad \text{(True)} \]
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\( b + c > a \): \[ 18 + 18 > 3 \] \[ 36 > 3 \quad \text{(True)} \]
Since all three conditions of the triangle inequality theorem are satisfied, a triangle can indeed be formed with sides of lengths 3, 18, and 18.