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
9, 4, 3
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Answer 1

To determine which group of side measures can form a triangle, we can apply the Triangle Inequality Theorem. The theorem states that for three sides \(a\), \(b\), and \(c\) (where \(a\), \(b\), and \(c\) are the lengths of the sides), the following conditions must be satisfied:

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

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

1. 2, 7, 6:

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

2. 12, 10, 22:

   • \(12 + 10 = 22 > 22\) (False)
   • Since this inequality does not hold, this group cannot form a triangle.

3. 8, 1, 7:

   • \(8 + 1 = 9 > 7\) (True)
   • \(8 + 7 = 15 > 1\) (True)
   • \(1 + 7 = 8 > 8\) (False)
   • Since one of the inequalities does not hold, this group cannot form a triangle.

4. 9, 4, 3:

   • \(9 + 4 = 13 > 3\) (True)
   • \(9 + 3 = 12 > 4\) (True)
   • \(4 + 3 = 7 > 9\) (False)
   • Since one of the inequalities does not hold, this group cannot form a triangle.

Summary:

The group of side measures that can form a triangle is 2, 7, 6.