Question

Apply the Triangle Inequality Theorem to determine whether sides with lengths of 5 inches, 10 inches, and 15 inches form a triangle.(1 point) Responses Yes, the majority of the conditions are met. Yes, the majority of the conditions are met. Yes, because the sum of 5+10 is less than or equal to 15. Yes, because the sum of 5 plus 10 is less than or equal to 15. No, because the sum of 5+10 is not greater than 15. No, because the sum of 5 plus 10 is not greater than 15. Yes, because the sum of 5+10 is not greater than 15.

Answer 1

To determine whether the sides 5 inches, 10 inches, and 15 inches can form a triangle, we apply the Triangle Inequality Theorem. The theorem states that for any three sides of a triangle, the following conditions must be met:

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

Let's denote the lengths as follows:
- \(a = 5\)
- \(b = 10\)
- \(c = 15\)

Now we check each condition:

1. \(a + b > c\):
\(5 + 10 > 15\)
\(15 > 15\) (This condition is **not satisfied**)

2. \(a + c > b\):
\(5 + 15 > 10\)
\(20 > 10\) (This condition is **satisfied**)

3. \(b + c > a\):
\(10 + 15 > 5\)
\(25 > 5\) (This condition is **satisfied**)

Since at least one of the conditions (specifically the first one) is not satisfied, the three lengths do not satisfy the Triangle Inequality Theorem. Therefore, the sides with lengths 5 inches, 10 inches, and 15 inches do not form a triangle.

The appropriate response is: **No, because the sum of 5+10 is not greater than 15.**