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

Question

Responses

8, 1, 7
8, 1, 7

9, 4, 3
9, 4, 3

2, 7, 6
2, 7, 6

12, 10, 22

Answer 1

To determine if a set of side measures can form a triangle, we apply the Triangle Inequality Theorem, which states that for any triangle with sides of lengths \( a \), \( b \), and \( c \), the following inequalities must hold:

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

Let's evaluate each set of side measures:

1. For the sides 8, 1, 7:

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

2. For the sides 9, 4, 3:

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

3. For the sides 2, 7, 6:

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

4. For the sides 12, 10, 22:

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

Based on the evaluations, the only group of side measures that will form a triangle is:

2, 7, 6.