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 which ones are the correct repsonses and choices

3,8,10
3,8,10

3,10,11
3,10,11

8,10,11
8,10,11

3,8,11

Answer 1

To determine which combinations of toothpick lengths can form triangles, we can apply the triangle inequality theorem. This theorem states that for any triangle with sides of lengths \(a\), \(b\), and \(c\) (where \(a \leq b \leq c\)), the following must hold true:

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

In practice, if the longest side is \(c\), we only need to check the first condition: \(a + b > c\).

Now, let's evaluate each combination:

1. 3, 8, 10

   • Longest side is 10.
   • Check: \(3 + 8 = 11 > 10\) (true)
   • Therefore, this combination can form a triangle.

2. 3, 10, 11

   • Longest side is 11.
   • Check: \(3 + 10 = 13 > 11\) (true)
   • Therefore, this combination can form a triangle.

3. 8, 10, 11

   • Longest side is 11.
   • Check: \(8 + 10 = 18 > 11\) (true)
   • Therefore, this combination can form a triangle.

4. 3, 8, 11

   • Longest side is 11.
   • Check: \(3 + 8 = 11 \not> 11\) (false)
   • Therefore, this combination cannot form a triangle.

Based on the evaluations:

• Combinations that can form triangles:
  • 3, 8, 10
  • 3, 10, 11
  • 8, 10, 11

Thus, the correct responses are:

• 3, 8, 10
• 3, 10, 11
• 8, 10, 11