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Joanna has four straws of different lengths. The table shows the lengths of the straws. Straw A B C D Length in inches 7 9 3 10Joanna makes as many triangles as she can using combinations of three of these straws. Determine how many different triangles she can make. What combinations of straws does she use? Show your work.(4 points)

5 answers

To determine how many different triangles Joanna can make with the straws, we need to apply the triangle inequality theorem. This theorem states that for any three lengths to form a triangle, the sum of any two lengths must be greater than the third length.

Joanna has the following straws:

  • Straw A: 7 inches
  • Straw B: 9 inches
  • Straw C: 3 inches
  • Straw D: 10 inches

Next, we can check all possible combinations of three straws to see which combinations can form a triangle.

Combinations of Straws:

  1. Straws A, B, C: (7, 9, 3)

    • Check:
      • \( 7 + 3 = 10 \) (not greater than 9)
      • \( 7 + 9 = 16 > 3 \)
      • \( 9 + 3 = 12 > 7 \)
    • Since 7 + 3 is not greater than 9, this combination does not form a triangle.

  2. Straws A, B, D: (7, 9, 10)

    • Check:
      • \( 7 + 9 = 16 > 10 \)
      • \( 7 + 10 = 17 > 9 \)
      • \( 9 + 10 = 19 > 7 \)
    • All sums are greater than the respective third lengths, so this combination forms a triangle.

  3. Straws A, C, D: (7, 3, 10)

    • Check:
      • \( 7 + 3 = 10 \) (not greater than 10)
      • \( 7 + 10 = 17 > 3 \)
      • \( 3 + 10 = 13 > 7 \)
    • Since 7 + 3 is not greater than 10, this combination does not form a triangle.

  4. Straws B, C, D: (9, 3, 10)

    • Check:
      • \( 9 + 3 = 12 > 10 \)
      • \( 9 + 10 = 19 > 3 \)
      • \( 3 + 10 = 13 > 9 \)
    • All sums are greater than the respective third lengths, so this combination forms a triangle.

Summary:

The valid combinations that can form triangles are:

  1. Straws A, B, D: (7, 9, 10)
  2. Straws B, C, D: (9, 3, 10)

Conclusion:

Joanna can make 2 different triangles using the following combinations of straws:

  1. Straw A, Straw B, Straw D
  2. Straw B, Straw C, Straw D

Thus, the answer is 2 triangles:

  • (7, 9, 10)
  • (9, 3, 10)
what about (7, 3, 9)?

Let's re-evaluate the combination of straws A, B, and C with lengths (7, 3, 9) to determine if they can form a triangle.

Combination A, B, C: (7, 3, 9)

We need to check if the triangle inequality holds for these three lengths:

  1. Check \( 7 + 3 > 9 \):

    • \( 7 + 3 = 10 \) which is greater than 9. (True)

  2. Check \( 7 + 9 > 3 \):

    • \( 7 + 9 = 16 \) which is greater than 3. (True)

  3. Check \( 3 + 9 > 7 \):

    • \( 3 + 9 = 12 \) which is greater than 7. (True)

Since all three conditions of the triangle inequality are satisfied, the combination (7, 3, 9) can indeed form a triangle.

Summary:

Now, let's summarize all valid combinations again:

  1. Straws A, B, C: (7, 3, 9) - Valid triangle
  2. Straws A, B, D: (7, 9, 10) - Valid triangle
  3. Straws B, C, D: (9, 3, 10) - Valid triangle

Final Conclusion:

Joanna can make 3 different triangles using the following combinations of straws:

  1. (7, 3, 9)
  2. (7, 9, 10)
  3. (9, 3, 10)

Thus, the corrected answer is 3 triangles. Thank you for pointing out the oversight!

No problem! Thanks for the help!

You're welcome! I'm glad I could assist you. If you have any more questions or need further help with anything else, feel free to ask. Have a great day!