Question
Joanna has four straws of different lengths. The table shows the lengths of the straws.%0D%0A%0D%0ALength of Straws%0D%0AStraw%09A%0D%0A𝐴%0D%0AB%0D%0A𝐵%0D%0AC%0D%0A𝐶%0D%0AD%0D%0A𝐷%0D%0ALength in inches%097%099%093%0910%0D%0AJoanna 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)
Answers
GPT-4o mini
To determine how many different triangles Joanna can make using combinations of the straws, we need to apply the triangle inequality theorem. This theorem states that for any three lengths \(a\), \(b\), and \(c\) to form a triangle, the following conditions must be met:
1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)
Given the lengths of the straws:
- Straw A = 7 inches
- Straw B = 9 inches
- Straw C = 3 inches
- Straw D = 10 inches
We will find the combinations of three straws and check if they satisfy the triangle inequality conditions.
**Step 1: List all combinations of three straws from A, B, C, and D.**
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)
**Step 2: Check each combination for the triangle inequality.**
1. **Combination A, B, C (7, 9, 3):**
- Check:
- \(7 + 9 = 16 > 3\) (True)
- \(7 + 3 = 10 > 9\) (False)
- \(9 + 3 = 12 > 7\) (True)
- This combination **cannot form a triangle**.
2. **Combination A, B, D (7, 9, 10):**
- Check:
- \(7 + 9 = 16 > 10\) (True)
- \(7 + 10 = 17 > 9\) (True)
- \(9 + 10 = 19 > 7\) (True)
- This combination **can form a triangle**.
3. **Combination A, C, D (7, 3, 10):**
- Check:
- \(7 + 3 = 10 > 10\) (False)
- \(7 + 10 = 17 > 3\) (True)
- \(3 + 10 = 13 > 7\) (True)
- This combination **cannot form a triangle**.
4. **Combination B, C, D (9, 3, 10):**
- Check:
- \(9 + 3 = 12 > 10\) (True)
- \(9 + 10 = 19 > 3\) (True)
- \(3 + 10 = 13 > 9\) (True)
- This combination **can form a triangle**.
**Step 3: Summary of valid triangles.**
- The valid combinations that can form triangles are:
- A, B, D (7, 9, 10)
- B, C, D (9, 3, 10)
Thus, Joanna can make **2 different triangles**:
1. Using straws A (7 inches), B (9 inches), and D (10 inches).
2. Using straws B (9 inches), C (3 inches), and D (10 inches).
**Final Answer:** Joanna can make **2 different triangles** with the combinations (A, B, D) and (B, C, D).
1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)
Given the lengths of the straws:
- Straw A = 7 inches
- Straw B = 9 inches
- Straw C = 3 inches
- Straw D = 10 inches
We will find the combinations of three straws and check if they satisfy the triangle inequality conditions.
**Step 1: List all combinations of three straws from A, B, C, and D.**
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)
**Step 2: Check each combination for the triangle inequality.**
1. **Combination A, B, C (7, 9, 3):**
- Check:
- \(7 + 9 = 16 > 3\) (True)
- \(7 + 3 = 10 > 9\) (False)
- \(9 + 3 = 12 > 7\) (True)
- This combination **cannot form a triangle**.
2. **Combination A, B, D (7, 9, 10):**
- Check:
- \(7 + 9 = 16 > 10\) (True)
- \(7 + 10 = 17 > 9\) (True)
- \(9 + 10 = 19 > 7\) (True)
- This combination **can form a triangle**.
3. **Combination A, C, D (7, 3, 10):**
- Check:
- \(7 + 3 = 10 > 10\) (False)
- \(7 + 10 = 17 > 3\) (True)
- \(3 + 10 = 13 > 7\) (True)
- This combination **cannot form a triangle**.
4. **Combination B, C, D (9, 3, 10):**
- Check:
- \(9 + 3 = 12 > 10\) (True)
- \(9 + 10 = 19 > 3\) (True)
- \(3 + 10 = 13 > 9\) (True)
- This combination **can form a triangle**.
**Step 3: Summary of valid triangles.**
- The valid combinations that can form triangles are:
- A, B, D (7, 9, 10)
- B, C, D (9, 3, 10)
Thus, Joanna can make **2 different triangles**:
1. Using straws A (7 inches), B (9 inches), and D (10 inches).
2. Using straws B (9 inches), C (3 inches), and D (10 inches).
**Final Answer:** Joanna can make **2 different triangles** with the combinations (A, B, D) and (B, C, D).