Sarah has toothpicks that are 4 different lengths, 3 in, 8in, 10in, 11in. She is trying to make as many triangles as she can. How many of the combinations below will make triangles according to the triangle inequality theorem? Pick ALL that apply(3 points)

Question

Responses

8,10,11
8,10,11

3,10,11
3,10,11

3,8,11
3,8,11

3,8,10

Answer 1

To determine if the combinations of toothpick lengths can form triangles, we can use the triangle inequality theorem. This theorem states that for any three lengths \(a\), \(b\), and \(c\) (where \(a \leq b \leq c\)), the following condition must hold:

\[ a + b > c \]

We'll analyze each combination:

1. Combination: 8, 10, 11

   • Check: \(8 + 10 = 18 > 11\) (True)
   • This combination can form a triangle.

2. Combination: 3, 10, 11

   • Check: \(3 + 10 = 13 > 11\) (True)
   • This combination can form a triangle.

3. Combination: 3, 8, 11

   • Check: \(3 + 8 = 11 > 11\) (False)
   • This combination cannot form a triangle.

4. Combination: 3, 8, 10

   • Check: \(3 + 8 = 11 > 10\) (True)
   • This combination can form a triangle.

So the combinations that can create triangles are:

• 8, 10, 11
• 3, 10, 11
• 3, 8, 10

The correct responses are:

• 8, 10, 11
• 3, 10, 11
• 3, 8, 10