Apply the Triangle Inequality Theorem and determine which group of side measures will form a triangle.(1 point)

Question

Responses

2, 7, 6
2, 7, 6

12, 10, 22
12, 10, 22

8, 1, 7
8, 1, 7

9, 4, 3

Answer 1

To determine whether a set of three side lengths 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 conditions must be met:

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

Let's apply this to each set of side measures provided:

1. 2, 7, 6

   • \(2 + 7 = 9 > 6\) (True)
   • \(2 + 6 = 8 > 7\) (True)
   • \(7 + 6 = 13 > 2\) (True)
   • Can form a triangle.

2. 12, 10, 22

   • \(12 + 10 = 22\) (Not greater than 22; fails this condition)
   • Cannot form a triangle.

3. 8, 1, 7

   • \(8 + 1 = 9 > 7\) (True)
   • \(8 + 7 = 15 > 1\) (True)
   • \(1 + 7 = 8 \text{ not greater than } 8\) (Fails this condition)
   • Cannot form a triangle.

4. 9, 4, 3

   • \(9 + 4 = 13 > 3\) (True)
   • \(9 + 3 = 12 > 4\) (True)
   • \(4 + 3 = 7 < 9\) (Fails this condition)
   • Cannot form a triangle.

In summary, only the side lengths 2, 7, 6 can form a triangle based on the Triangle Inequality Theorem.