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

Question

Responses

12, 10, 22
12, 10, 22

8, 1, 7
8, 1, 7

9, 4, 3
9, 4, 3

2, 7, 6

Answer 1

The Triangle Inequality Theorem states that for any three sides \( a \), \( b \), and \( c \) of a triangle, the following inequalities must hold true:

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

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

1. 12, 10, 22

   • \( 12 + 10 = 22 \) (not greater)
   • \( 12 + 22 = 34 > 10 \)
   • \( 10 + 22 = 32 > 12 \)
   • Does not form a triangle.

2. 8, 1, 7

   • \( 8 + 1 = 9 > 7 \)
   • \( 8 + 7 = 15 > 1 \)
   • \( 1 + 7 = 8 \) (not greater)
   • Does not form a triangle.

3. 9, 4, 3

   • \( 9 + 4 = 13 > 3 \)
   • \( 9 + 3 = 12 > 4 \)
   • \( 4 + 3 = 7 < 9 \) (not greater)
   • Does not form a triangle.

4. 2, 7, 6

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

Based on the Triangle Inequality Theorem, the only group of side measures that will form a triangle is 2, 7, 6.