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

3,10,11
3,10,11

3,8,10
3,8,10

3,8,11
3,8,11

8,10,11

Answer 1

To determine which combinations of toothpick lengths can form a triangle, we can apply the triangle inequality theorem. The theorem states that for three 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. \( b + c > a \)

However, if \( a + b > c\) is satisfied, then the other two inequalities will also be satisfied, so we generally only need to check the first inequality for each combination.

Let's analyze each of the options:

1. 3, 10, 11

   • \( 3 + 10 = 13 > 11 \) (satisfied)
   • Therefore, this combination can form a triangle.

2. 3, 8, 10

   • \( 3 + 8 = 11 > 10 \) (satisfied)
   • Therefore, this combination can form a triangle.

3. 3, 8, 11

   • \( 3 + 8 = 11 \not> 11 \) (not satisfied)
   • Therefore, this combination cannot form a triangle.

4. 8, 10, 11

   • \( 8 + 10 = 18 > 11 \) (satisfied)
   • Therefore, this combination can form a triangle.

Based on the analysis, the combinations that can form triangles are:

• 3, 10, 11
• 3, 8, 10
• 8, 10, 11

The only combination that cannot form a triangle is:

• 3, 8, 11

Thus, the combinations that will make triangles according to the triangle inequality theorem are:

• 3, 10, 11
• 3, 8, 10
• 8, 10, 11