Apply the Triangle Inequality Theorem and determine which group of side measures will form a triangle. Responses: 9, 4, 3 9, 4, 3 2, 7, 6 2, 7, 6 12, 10, 22 12, 10, 22 8, 1, 7

Answer 1

To determine whether a set of side measures can form a triangle, we can use the Triangle Inequality Theorem. According to the theorem, for any three side lengths \( a, b, c \), the following must hold true:

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

Let's check each of the groups of side measures one by one:

1. For the group \( 9, 4, 3 \):

   • \( 9 + 4 = 13 > 3 \) (True)
   • \( 9 + 3 = 12 > 4 \) (True)
   • \( 4 + 3 = 7 > 9 \) (False)
   • This group does NOT form a triangle.

2. For the group \( 2, 7, 6 \):

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

3. For the group \( 12, 10, 22 \):

   • \( 12 + 10 = 22 \not> 22 \) (False)
   • This group does NOT form a triangle.

4. For the group \( 8, 1, 7 \):

   • \( 8 + 1 = 9 > 7 \) (True)
   • \( 8 + 7 = 15 > 1 \) (True)
   • \( 1 + 7 = 8 \not> 8 \) (False)
   • This group does NOT form a triangle.

Summary:

• \( 9, 4, 3 \): Does not form a triangle
• \( 2, 7, 6 \): Forms a triangle
• \( 12, 10, 22 \): Does not form a triangle
• \( 8, 1, 7 \): Does not form a triangle