To determine whether Sarah can make a triangle with combinations of toothpicks of different lengths, we can use the triangle inequality theorem. According to this theorem, for any triangle with side lengths \( a \), \( b \), and \( c \), the following must hold true:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Given the toothpick lengths: 3 inches, 8 inches, 10 inches, and 11 inches, we can check combinations of 3 lengths at a time to see if they can form a triangle.
Let's evaluate all the combinations:
-
3, 8, 10:
- \( 3 + 8 = 11 \) (not > 10)
- This combination cannot form a triangle.
-
3, 8, 11:
- \( 3 + 8 = 11 \) (not > 11)
- This combination cannot form a triangle.
-
3, 10, 11:
- \( 3 + 10 = 13 > 11 \) (holds)
- \( 3 + 11 = 14 > 10 \) (holds)
- \( 10 + 11 = 21 > 3 \) (holds)
- This combination can form a triangle.
-
8, 10, 11:
- \( 8 + 10 = 18 > 11 \) (holds)
- \( 8 + 11 = 19 > 10 \) (holds)
- \( 10 + 11 = 21 > 8 \) (holds)
- This combination can form a triangle.
So, based on the evaluations:
- The valid combinations that can form triangles are (3, 10, 11) and (8, 10, 11).
- Hence, Sarah can make 2 triangles with the toothpicks of given lengths.