To determine if the sides of lengths 2, 7, and 8 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 be satisfied:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
Assigning the lengths as follows:
- \(a = 2\)
- \(b = 7\)
- \(c = 8\)
Now, we can test the triangle inequalities:
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\(a + b > c\): \[ 2 + 7 > 8 \quad \text{(This simplifies to } 9 > 8\text{, which is true)} \]
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\(a + c > b\): \[ 2 + 8 > 7 \quad \text{(This simplifies to } 10 > 7\text{, which is true)} \]
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\(b + c > a\): \[ 7 + 8 > 2 \quad \text{(This simplifies to } 15 > 2\text{, which is true)} \]
Since all three inequalities are satisfied, the sides of lengths 2, 7, and 8 can indeed form a triangle.
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