Question

Can the sides of a triangle have lengths 8, 8, and 16?

Answer 1

To determine if the sides of a triangle with lengths 8, 8, and 16 can form a valid triangle, we can apply the triangle inequality theorem. The triangle inequality states that for any triangle with sides of lengths \(a\), \(b\), and \(c\):

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

In this case, we can assign \(a = 8\), \(b = 8\), and \(c = 16\).

Now, we can check each of the inequalities:

1. \(a + b > c\):
\[
8 + 8 > 16 \quad \Rightarrow \quad 16 > 16 \quad \text{(not true)}
\]

2. \(a + c > b\):
\[
8 + 16 > 8 \quad \Rightarrow \quad 24 > 8 \quad \text{(true)}
\]

3. \(b + c > a\):
\[
8 + 16 > 8 \quad \Rightarrow \quad 24 > 8 \quad \text{(true)}
\]

Since the first inequality \(8 + 8 > 16\) is not true, the lengths 8, 8, and 16 do not satisfy the triangle inequality theorem. Therefore, these sides cannot form a triangle.