To determine whether the sides of lengths 3, 9, and 9 can form a triangle, we can use the triangle inequality theorem. This theorem states that for any triangle with sides of lengths \( a \), \( b \), and \( c \), the following conditions must hold:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Let's apply these conditions to our side lengths \( 3 \), \( 9 \), and \( 9 \).
Let \( a = 3 \), \( b = 9 \), and \( c = 9 \).
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Check \( a + b > c \): \[ 3 + 9 > 9 \quad \Rightarrow \quad 12 > 9 \quad \text{(True)} \]
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Check \( a + c > b \): \[ 3 + 9 > 9 \quad \Rightarrow \quad 12 > 9 \quad \text{(True)} \]
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Check \( b + c > a \): \[ 9 + 9 > 3 \quad \Rightarrow \quad 18 > 3 \quad \text{(True)} \]
Since all three conditions of the triangle inequality are satisfied, the sides of lengths 3, 9, and 9 can indeed form a triangle.