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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, 3,10 Joanna 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) Math Short Answer Rubric (2 points)Points AwardedCriteria2The answer is correct. The student made no errors while solving.The answer shows that the student used problem-solving skills and reasoning.All work is shown or explained.1The answer is partially correct. The student made errors while solving.The answer shows little problem-solving skills or reasoning.Some work is shown or explained.0The question is not answered. Skip to navigation

1 answer

To determine how many different triangles Joanna can make using combinations of three of the four straws, we can follow the triangle inequality theorem. According to this theorem, for any three side lengths \(a, b, c\) to form a triangle, the following conditions must be satisfied:

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

We will denote the lengths of the straws as follows:

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

First, we will examine all the combinations of three straws from the four available. The combinations are:

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

Next, we will evaluate each combination against the triangle inequality theorem.

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

    • \(7 + 9 > 3\) → \(16 > 3\) (True)
    • \(7 + 3 > 9\) → \(10 > 9\) (True)
    • \(9 + 3 > 7\) → \(12 > 7\) (True)

    This combination can form a triangle.

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

    • \(7 + 9 > 10\) → \(16 > 10\) (True)
    • \(7 + 10 > 9\) → \(17 > 9\) (True)
    • \(9 + 10 > 7\) → \(19 > 7\) (True)

    This combination can also form a triangle.

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

    • \(7 + 3 > 10\) → \(10 > 10\) (False; does not satisfy)
    • \(7 + 10 > 3\) → \(17 > 3\) (True)
    • \(3 + 10 > 7\) → \(13 > 7\) (True)

    This combination cannot form a triangle.

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

    • \(9 + 3 > 10\) → \(12 > 10\) (True)
    • \(9 + 10 > 3\) → \(19 > 3\) (True)
    • \(3 + 10 > 9\) → \(13 > 9\) (True)

    This combination can also form a triangle.

In summary, the valid combinations that can form a triangle are:

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

Therefore, Joanna can make a total of 3 different triangles using the straws.

Final answer: Joanna can make 3 different triangles using the combinations of straws (7, 9, 3), (7, 9, 10), and (9, 3, 10).