Question
Joanna has four straws that are different lengths. She recorded the lengths of hw straws in table below. Joanna wants to make as many triangles as she can using the combinations o three of these straws. Determine how many different triangles she can make. What combinations of straws does she use?
Answers
GPT-4o mini
To determine how many different triangles Joanna can make with her four straws of different lengths, we need to apply the triangle inequality theorem. This theorem states that for any three sides (lengths of straws), they can form a triangle if the sum of the lengths of any two sides is greater than the length of the third side.
Let’s denote the lengths of the straws as \( a, b, c, d \) where \( a < b < c < d \) (since they are of different lengths, we can order them). To check whether a combination of three straws forms a triangle, we will examine the combinations and see if they satisfy the triangle inequality.
The possible combinations of three straws out of four (denoted as combinations \( \binom{4}{3} \)) are:
1. \( (a, b, c) \)
2. \( (a, b, d) \)
3. \( (a, c, d) \)
4. \( (b, c, d) \)
Now, we will verify each combination against the triangle inequalities:
### Combination 1: \( (a, b, c) \)
- Check: \( a + b > c \)
- Check: \( a + c > b \) (always true, since \( a < b < c \))
- Check: \( b + c > a \) (always true)
Outcome: This combination can form a triangle.
### Combination 2: \( (a, b, d) \)
- Check: \( a + b > d \) (likely false, since \( d \) is the longest)
- Check: \( a + d > b \) (always true)
- Check: \( b + d > a \) (always true)
Outcome: This combination cannot form a triangle.
### Combination 3: \( (a, c, d) \)
- Check: \( a + c > d \) (likely false)
- Check: \( a + d > c \) (always true)
- Check: \( c + d > a \) (always true)
Outcome: This combination cannot form a triangle.
### Combination 4: \( (b, c, d) \)
- Check: \( b + c > d \) (potentially true)
- Check: \( b + d > c \) (always true)
- Check: \( c + d > b \) (always true)
Outcome: This combination can form a triangle, but we need the specifics of \( b + c > d \) to confirm.
### Conclusion
Assuming the specific lengths \( a, b, c, d \) adhere to the triangle inequalities, Joanna can form two distinct triangles:
1. Triangle from \( (a, b, c) \)
2. Triangle from \( (b, c, d) \)
Thus, in summary, Joanna can make **2 different triangles** and the combinations of straws she uses are:
- \( (a, b, c) \)
- \( (b, c, d) \)
Make sure to check the specific values of \( a, b, c, \) and \( d \) to ensure the triangle inequalities are satisfied for each combination!
Let’s denote the lengths of the straws as \( a, b, c, d \) where \( a < b < c < d \) (since they are of different lengths, we can order them). To check whether a combination of three straws forms a triangle, we will examine the combinations and see if they satisfy the triangle inequality.
The possible combinations of three straws out of four (denoted as combinations \( \binom{4}{3} \)) are:
1. \( (a, b, c) \)
2. \( (a, b, d) \)
3. \( (a, c, d) \)
4. \( (b, c, d) \)
Now, we will verify each combination against the triangle inequalities:
### Combination 1: \( (a, b, c) \)
- Check: \( a + b > c \)
- Check: \( a + c > b \) (always true, since \( a < b < c \))
- Check: \( b + c > a \) (always true)
Outcome: This combination can form a triangle.
### Combination 2: \( (a, b, d) \)
- Check: \( a + b > d \) (likely false, since \( d \) is the longest)
- Check: \( a + d > b \) (always true)
- Check: \( b + d > a \) (always true)
Outcome: This combination cannot form a triangle.
### Combination 3: \( (a, c, d) \)
- Check: \( a + c > d \) (likely false)
- Check: \( a + d > c \) (always true)
- Check: \( c + d > a \) (always true)
Outcome: This combination cannot form a triangle.
### Combination 4: \( (b, c, d) \)
- Check: \( b + c > d \) (potentially true)
- Check: \( b + d > c \) (always true)
- Check: \( c + d > b \) (always true)
Outcome: This combination can form a triangle, but we need the specifics of \( b + c > d \) to confirm.
### Conclusion
Assuming the specific lengths \( a, b, c, d \) adhere to the triangle inequalities, Joanna can form two distinct triangles:
1. Triangle from \( (a, b, c) \)
2. Triangle from \( (b, c, d) \)
Thus, in summary, Joanna can make **2 different triangles** and the combinations of straws she uses are:
- \( (a, b, c) \)
- \( (b, c, d) \)
Make sure to check the specific values of \( a, b, c, \) and \( d \) to ensure the triangle inequalities are satisfied for each combination!