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

3,8,10
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 a triangle, we can use the triangle inequality theorem. The theorem states that for any triangle with side lengths a, b, and c, the following conditions must hold:

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

Let's evaluate each of the combinations you provided:

1. 3, 8, 10:

   • \(3 + 8 = 11 > 10\) (satisfied)
   • \(3 + 10 = 13 > 8\) (satisfied)
   • \(8 + 10 = 18 > 3\) (satisfied)
   • This combination forms a triangle (valid).

2. 3, 10, 11:

   • \(3 + 10 = 13 > 11\) (satisfied)
   • \(3 + 11 = 14 > 10\) (satisfied)
   • \(10 + 11 = 21 > 3\) (satisfied)
   • This combination forms a triangle (valid).

3. 8, 10, 11:

   • \(8 + 10 = 18 > 11\) (satisfied)
   • \(8 + 11 = 19 > 10\) (satisfied)
   • \(10 + 11 = 21 > 8\) (satisfied)
   • This combination forms a triangle (valid).

4. 3, 8, 11:

   • \(3 + 8 = 11\) (not greater than 11, fails)
   • \(3 + 11 = 14 > 8\) (satisfied)
   • \(8 + 11 = 19 > 3\) (satisfied)
   • Since one condition fails, this combination does not form a triangle (invalid).

Now, summarizing the results:

• Valid combinations:

  • 3, 8, 10
  • 3, 10, 11
  • 8, 10, 11

• Invalid combination:

  • 3, 8, 11

So the combinations that will make triangles according to the triangle inequality theorem are: 3, 8, 10; 3, 10, 11; 8, 10, 11.