Question

or
Can the sides of a triangle have lengths 2, 7, and 8?

Answer 1

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:

1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

Assigning the lengths as follows:
- \(a = 2\)
- \(b = 7\)
- \(c = 8\)

Now, we can test the triangle inequalities:

1. \(a + b > c\):
\[
2 + 7 > 8 \quad \text{(This simplifies to } 9 > 8\text{, which is true)}
\]

2. \(a + c > b\):
\[
2 + 8 > 7 \quad \text{(This simplifies to } 10 > 7\text{, which is true)}
\]

3. \(b + c > a\):
\[
7 + 8 > 2 \quad \text{(This simplifies to } 15 > 2\text{, which is true)}
\]

Since all three inequalities are satisfied, the sides of lengths 2, 7, and 8 can indeed form a triangle.