To determine whether the lengths of 4 inches, 8 inches, and 9 inches can form a triangle, we can 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:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
Let's denote the sides as follows:
- \(a = 4\) inches
- \(b = 8\) inches
- \(c = 9\) inches
Now we can check each of the inequalities:
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Check \(a + b > c\): \[ 4 + 8 > 9 \implies 12 > 9 \quad \text{(True)} \]
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Check \(a + c > b\): \[ 4 + 9 > 8 \implies 13 > 8 \quad \text{(True)} \]
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Check \(b + c > a\): \[ 8 + 9 > 4 \implies 17 > 4 \quad \text{(True)} \]
Since all three inequalities are satisfied, the sides with lengths of 4 inches, 8 inches, and 9 inches can indeed form a triangle.