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 using the triangle inequality theorem.
To form a triangle using three straws, the sum of the lengths of any two sides must be greater than the length of the remaining side.
Let's analyze the options:
1. A, B, C
The length of A + B = 7 + 9 = 16, which is greater than the length of C (3).
The length of A + C = 7 + 3 = 10, which is not greater than the length of B (9).
The length of B + C = 9 + 3 = 12, which is greater than the length of A (7).
2. A, B, D
The length of A + B = 7 + 9 = 16, which is greater than the length of D (10).
The length of A + D = 7 + 10 = 17, which is greater than the length of B (9).
The length of B + D = 9 + 10 = 19, which is greater than the length of A (7).
3. A, C, D
The length of A + C = 7 + 3 = 10, which is not greater than the length of D (10).
The length of A + D = 7 + 10 = 17, which is greater than the length of C (3).
The length of C + D = 3 + 10 = 13, which is greater than the length of A (7).
4. B, C, D
The length of B + C = 9 + 3 = 12, which is greater than the length of D (10).
The length of B + D = 9 + 10 = 19, which is greater than the length of C (3).
The length of C + D = 3 + 10 = 13, which is greater than the length of B (9).
Based on the triangle inequality theorem, Joanna can only form triangles using straws A, B, and D. Therefore, there is only one combination of straws she can use to form a triangle: A, B, and D.
To determine how many different triangles Joanna can make, we need to use the triangle inequality theorem. According to the theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's evaluate the triangle possibilities using the given lengths:
1. For straw A (length = 7):
- A, B, C: 7 + 9 > 3 (True)
- A, B, D: 7 + 10 > 9 (True)
- A, C, D: 7 + 3 > 10 (False) -- Not a valid triangle.
2. For straw B (length = 9):
- B, A, C: 9 + 7 > 3 (True)
- B, A, D: 9 + 10 > 7 (True)
- B, C, D: 9 + 3 > 10 (False) -- Not a valid triangle.
3. For straw C (length = 3):
- C, A, B: 3 + 7 > 9 (False) -- Not a valid triangle.
4. For straw D (length = 10):
- D, A, B: 10 + 7 > 9 (True)
- D, A, C: 10 + 3 > 7 (True)
- D, B, C: 10 + 3 > 9 (True)
Based on the triangle inequality theorem, Joanna can make the following triangles:
- A, B, C
- A, B, D
- D, A, B
- D, A, C
- D, B, C
Therefore, Joanna can make a total of 5 different triangles using combinations of three straws.