Can the sides of a triangle have lengths 3, 7, and 9?

Answer 1

To determine if three lengths can form a triangle, we need to 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. \(b + c > a\)

In this case, we have the side lengths 3, 7, and 9. Let's assign:

• \(a = 3\)
• \(b = 7\)
• \(c = 9\)

Now, we will check the inequalities:

1. \(3 + 7 > 9\)

   • \(10 > 9\) (True)

2. \(3 + 9 > 7\)

   • \(12 > 7\) (True)

3. \(7 + 9 > 3\)

   • \(16 > 3\) (True)

Since all three conditions of the triangle inequality are satisfied, the lengths 3, 7, and 9 can indeed form a triangle.