Joanna has four straws of different lengths. The table shows the lengths of the straws.

Question

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)

Answer 1

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

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

Where \(a\), \(b\), and \(c\) are the lengths of the straws used.

The lengths of the straws are:

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

Now, we will evaluate the combinations of three straws:

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

   • \(7 + 9 = 16 > 3\) (satisfied)
   • \(7 + 3 = 10 > 9\) (not satisfied)
   • Cannot form a triangle.

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

   • \(7 + 9 = 16 > 10\) (satisfied)
   • \(7 + 10 = 17 > 9\) (satisfied)
   • \(9 + 10 = 19 > 7\) (satisfied)
   • Can form a triangle.

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

   • \(7 + 3 = 10 \not> 10\) (not satisfied)
   • Cannot form a triangle.

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

   • \(9 + 3 = 12 > 10\) (satisfied)
   • \(9 + 10 = 19 > 3\) (satisfied)
   • \(3 + 10 = 13 > 9\) (satisfied)
   • Can form a triangle.

Now we summarize the findings:

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

Thus, Joanna can make 2 different triangles. The combinations of straws used to form triangles are:

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