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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

1 answer

To determine how many different triangles Joanna can make with her straws, we will use the triangle inequality theorem. According to this theorem, three lengths \(a\), \(b\), and \(c\) can form a triangle if the following conditions are satisfied:

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

First, let's list the lengths of the straws:

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

Next, we'll consider all combinations of three straws and apply the triangle inequality to each combination.

The combinations of the straws can be represented as follows:

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

Now, let's check each combination:

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

    • Check inequalities:
      • \(7 + 9 > 3\) → \(16 > 3\) (True)
      • \(7 + 3 > 9\) → \(10 > 9\) (True)
      • \(9 + 3 > 7\) → \(12 > 7\) (True)
    • All inequalities hold; thus, this combination can form a triangle.

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

    • Check inequalities:
      • \(7 + 9 > 10\) → \(16 > 10\) (True)
      • \(7 + 10 > 9\) → \(17 > 9\) (True)
      • \(9 + 10 > 7\) → \(19 > 7\) (True)
    • All inequalities hold; thus, this combination can form a triangle.

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

    • Check inequalities:
      • \(7 + 3 > 10\) → \(10 > 10\) (False, as it is not greater)
      • \(7 + 10 > 3\) → \(17 > 3\) (True)
      • \(3 + 10 > 7\) → \(13 > 7\) (True)
    • Since one inequality fails, this combination cannot form a triangle.

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

    • Check inequalities:
      • \(9 + 3 > 10\) → \(12 > 10\) (True)
      • \(9 + 10 > 3\) → \(19 > 3\) (True)
      • \(3 + 10 > 9\) → \(13 > 9\) (True)
    • All inequalities hold; thus, this combination can form a triangle.

Summary of Triangle Combinations:

  • The combinations that can form triangles are:
    • A, B, C (7, 9, 3)
    • A, B, D (7, 9, 10)
    • B, C, D (9, 3, 10)

Total number of different triangles: 3

Therefore, Joanna can make 3 different triangles with the following straw combinations:

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