Question

Triangle inequality theorem quick check
Apply the triangle inequality theorem to determine whether sides with lengths of 4.24 inches, 9.89 inches, and 8.75 inches form a triangle

Answer 1

To determine whether the sides of lengths 4.24 inches, 9.89 inches, and 8.75 inches can form a triangle, we can apply the triangle inequality theorem. According to this theorem, for three sides \(a\), \(b\), and \(c\), the following conditions must be satisfied:

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

Let's denote the sides as follows:
- \(a = 4.24\) inches
- \(b = 9.89\) inches
- \(c = 8.75\) inches

Now, we'll check each of the inequalities:

1. \(a + b > c\):
\[
4.24 + 9.89 = 14.13 > 8.75 \quad \text{(True)}
\]

2. \(a + c > b\):
\[
4.24 + 8.75 = 13.00 > 9.89 \quad \text{(True)}
\]

3. \(b + c > a\):
\[
9.89 + 8.75 = 18.64 > 4.24 \quad \text{(True)}
\]

Since all three inequalities are satisfied, the sides with lengths 4.24 inches, 9.89 inches, and 8.75 inches can indeed form a triangle.