Sarah has toothpicks that are 4 different lengths, 3in, 8in, 10in, 11in. She is trying to make as many triangles as she can. How many of the combination will make triangle according to the triangle inequality theorem?

Answer 1

To determine how many combinations of toothpicks can form a triangle according to the triangle inequality theorem, we can follow these steps:

The triangle inequality theorem states that for any triangle with sides of lengths \(a\), \(b\), and \(c\), the following inequalities must hold:

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

For toothpicks of lengths 3 inches, 8 inches, 10 inches, and 11 inches, we will check all possible combinations of three lengths.

The combinations of three lengths from the four available lengths are:

1. \(3, 8, 10\)
2. \(3, 8, 11\)
3. \(3, 10, 11\)
4. \(8, 10, 11\)

Now, we will check each combination for the triangle inequalities.

1. Combination \(3, 8, 10\)

• \(3 + 8 > 10 \rightarrow 11 > 10\) (True)
• \(3 + 10 > 8 \rightarrow 13 > 8\) (True)
• \(8 + 10 > 3 \rightarrow 18 > 3\) (True)

This combination can form a triangle.

2. Combination \(3, 8, 11\)

• \(3 + 8 > 11 \rightarrow 11 > 11\) (False)
• \(3 + 11 > 8 \rightarrow 14 > 8\) (True)
• \(8 + 11 > 3 \rightarrow 19 > 3\) (True)

This combination cannot form a triangle.

3. Combination \(3, 10, 11\)

• \(3 + 10 > 11 \rightarrow 13 > 11\) (True)
• \(3 + 11 > 10 \rightarrow 14 > 10\) (True)
• \(10 + 11 > 3 \rightarrow 21 > 3\) (True)

This combination can form a triangle.

4. Combination \(8, 10, 11\)

• \(8 + 10 > 11 \rightarrow 18 > 11\) (True)
• \(8 + 11 > 10 \rightarrow 19 > 10\) (True)
• \(10 + 11 > 8 \rightarrow 21 > 8\) (True)

This combination can form a triangle.

Conclusion:

The combinations that can form triangles are:

1. \(3, 8, 10\)
2. \(3, 10, 11\)
3. \(8, 10, 11\)

Thus, the total number of combinations that can form triangles is 3.