Question

Can three side lengths of 7 inches, 8 inches and 9 inches form zero, one, or two triangles?

Answer 1

To determine whether the side lengths of 7 inches, 8 inches, and 9 inches can form triangles, we apply the triangle inequality theorem. The triangle inequality states that for any triangle with sides of lengths \(a\), \(b\), and \(c\), the following must hold true:

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

Let's denote the sides as follows:
- \(a = 7\)
- \(b = 8\)
- \(c = 9\)

Now, we can check the triangle inequalities:

1. \(7 + 8 > 9\):
\[
15 > 9 \quad \text{(True)}
\]

2. \(7 + 9 > 8\):
\[
16 > 8 \quad \text{(True)}
\]

3. \(8 + 9 > 7\):
\[
17 > 7 \quad \text{(True)}
\]

Since all three triangle inequalities hold true, we conclude that these side lengths can form a triangle.

Furthermore, with any three side lengths that satisfy the triangle inequalities, only one unique triangle can be formed with those specific lengths, unless the sides can be rotated or reflected, but that does not count as distinct triangles.

Thus, the side lengths of 7 inches, 8 inches, and 9 inches can form **exactly one triangle**.