Question

Apply the Triangle Inequality Theorem to determine whether sides with lengths of 3 inches, 4 inches, and 9 inches form a triangle.(1 point) Responses Yes, because the sum of any two sides is less than the third side. Yes, because the sum of any two sides is less than the third side. Yes, the majority of the conditions are met. Yes, the majority of the conditions are met. Yes, because 3+4 is less than 9. Yes, because 3 plus 4 is less than 9. No, because 3+4 is less than 9.

Answer 1

To determine whether sides of lengths 3 inches, 4 inches, and 9 inches can form a triangle, we can apply the Triangle Inequality Theorem. According to this theorem, for any 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 assign \(a = 3\), \(b = 4\), and \(c = 9\) and check the conditions:

1. \(3 + 4 > 9\) (This is false, as 7 is not greater than 9)
2. \(3 + 9 > 4\) (This is true, as 12 is greater than 4)
3. \(4 + 9 > 3\) (This is true, as 13 is greater than 3)

Since the first condition fails (3 + 4 is not greater than 9), the sides cannot form a triangle.

The correct response is:

**No, because 3+4 is less than 9.**