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jonana 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)

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To determine how many different triangles Joanna can make using the straws of lengths 3, 7, 9, and 10 inches, we can use the triangle inequality theorem. This theorem states that for three sides to form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

Step 1: List all combinations of three straws

The possible combinations of three straws (choosing from straw lengths 3, 7, 9, and 10 inches) are:

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

Step 2: Check each combination using the triangle inequality theorem

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

    • 7 + 3 > 9? → 10 > 9 (True)
    • 7 + 9 > 3? → 16 > 3 (True)
    • 3 + 9 > 7? → 12 > 7 (True)
    • Valid triangle

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

    • 7 + 9 > 10? → 16 > 10 (True)
    • 7 + 10 > 9? → 17 > 9 (True)
    • 9 + 10 > 7? → 19 > 7 (True)
    • Valid triangle

  3. Combination A, C, D (Lengths: 7, 3, 10):

    • 7 + 3 > 10? → 10 > 10 (False)
    • Not a valid triangle

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

    • 9 + 3 > 10? → 12 > 10 (True)
    • 9 + 10 > 3? → 19 > 3 (True)
    • 3 + 10 > 9? → 13 > 9 (True)
    • Valid triangle

Step 3: Summary of valid triangles

The valid triangles are:

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

Conclusion

In total, Joanna can form 3 different triangles using the combinations above. The combinations and lengths used are:

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