Question

Triangle inequality theorem quick check
Apply the triangle inequality theorem to determine whether sides with lengths of 4.24 inches, 9.89 inches, and 8.75 inches form a triangle
No, 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 the sum of any two sides is greater than the third side
Yes, because the sum of any two sides is greater than the third side

Answer 1

To determine if the sides with lengths of 4.24 inches, 9.89 inches, and 8.75 inches can form a triangle, we apply the triangle inequality theorem. The theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

Let's denote the sides as follows:
- \( a = 4.24 \)
- \( b = 9.89 \)
- \( c = 8.75 \)

We need to check the following three inequalities:

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

Now, let's calculate each condition:

1. \( a + b = 4.24 + 9.89 = 14.13 \) and \( c = 8.75 \):
- \( 14.13 > 8.75 \) (True)

2. \( a + c = 4.24 + 8.75 = 13.00 \) and \( b = 9.89 \):
- \( 13.00 > 9.89 \) (True)

3. \( b + c = 9.89 + 8.75 = 18.64 \) and \( a = 4.24 \):
- \( 18.64 > 4.24 \) (True)

Since all three conditions are satisfied, the sides can indeed form a triangle.

Thus, the correct answer is: **Yes, because the sum of any two sides is greater than the third side.**