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

Question

Responses

9, 4, 3
9, 4, 3

2, 7, 6
2, 7, 6

12, 10, 22
12, 10, 22

8, 1, 7
8, 1, 7

Answer 1

To determine if a group of side measures 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 all be true:

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

Let's apply this to each group of side measures:

1. 9, 4, 3

   • \( 9 + 4 = 13 > 3 \) (True)
   • \( 9 + 3 = 12 > 4 \) (True)
   • \( 4 + 3 = 7 > 9 \) (False)
   • Conclusion: Cannot form a triangle.

2. 2, 7, 6

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

3. 12, 10, 22

   • \( 12 + 10 = 22 > 22 \) (False; must be greater, not equal)
   • Conclusion: Cannot form a triangle.

4. 8, 1, 7

   • \( 8 + 1 = 9 > 7 \) (True)
   • \( 8 + 7 = 15 > 1 \) (True)
   • \( 1 + 7 = 8 > 8 \) (False; must be greater, not equal)
   • Conclusion: Cannot form a triangle.

Based on the Triangle Inequality Theorem, only the group 2, 7, 6 can form a triangle.