Apply the Triangle Inequality Theorem to determine whether sides with lengths of 3 inches, 4 inches, and 9 inches form a triangle.(1 point) Responses Yes, because the sum of any two sides is less than the third side. Yes, because the sum of any two sides is less than the third side. No, because 3+4 is less than 9. No, because 3 plus 4 is less than 9. Yes, the majority of the conditions are met. Yes, the majority of the conditions are met. Yes, because 3+4 is less than 9.

Answer 1

To determine if the sides with lengths of 3 inches, 4 inches, and 9 inches can form a triangle, we apply the Triangle Inequality Theorem. According to this theorem, for any three sides \( a \), \( b \), and \( c \) of a triangle, the following conditions must be met:

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

Let's label the sides:

• \( a = 3 \)
• \( b = 4 \)
• \( c = 9 \)

Now, we check each condition:

1. \( 3 + 4 > 9 \) → \( 7 > 9 \) (False)
2. \( 3 + 9 > 4 \) → \( 12 > 4 \) (True)
3. \( 4 + 9 > 3 \) → \( 13 > 3 \) (True)

The first condition (3 + 4 > 9) is false. Since one of the conditions of the Triangle Inequality Theorem is not satisfied, the three lengths do not form a triangle.

Therefore, the correct response is: No, because 3 + 4 is less than 9.