Question

Apply the Triangle Inequality Theorem to determine whether sides with lengths of 4 inches, 8 inches, and 9 inches form a triangle.(1 point)
Responses

Answer 1

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:

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

Let's denote the sides as follows:
- \(a = 4\) inches
- \(b = 8\) inches
- \(c = 9\) inches

Now we can check each of the inequalities:

1. **Check \(a + b > c\)**:
\[
4 + 8 > 9 \implies 12 > 9 \quad \text{(True)}
\]

2. **Check \(a + c > b\)**:
\[
4 + 9 > 8 \implies 13 > 8 \quad \text{(True)}
\]

3. **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.