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.
To form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
Let's try to make triangles using each straw as the longest side:
1) A as the longest side: B + C > A (9 + 3 > 7) - This is true, so A, B, and C can form a triangle.
2) A as the longest side: B + D > A (9 + 10 > 7) - This is true, so A, B, and D can form a triangle.
3) A as the longest side: C + D > A (3 + 10 > 7) - This is true, so A, C, and D can form a triangle.
Next, let's try B as the longest side:
4) B as the longest side: A + C > B (7 + 3 > 9) - This is true, so B, A, and C can form a triangle.
5) B as the longest side: A + D > B (7 + 10 > 9) - This is true, so B, A, and D can form a triangle.
Finally, let's try C as the longest side:
6) C as the longest side: A + B > C (7 + 9 > 3) - This is true, so C, A, and B can form a triangle.
Therefore, Joanna can form 6 different triangles using combinations of three straws:
1) (A, B, C)
2) (A, B, D)
3) (A, C, D)
4) (B, A, C)
5) (B, A, D)
6) (C, A, B)
Well, let's get cracking with straws and triangles! But first, let's see how many combinations we can make using three straws out of the four.
Since Joanna has four straws (A, B, C, D), we can use the combination formula to determine how many possible combinations there are. The formula is 𝑛! / (𝑟! (𝑛 − 𝑟)!), where 𝑛 is the total number of items and 𝑟 is how many items we want in each combination.
Using this formula, we can calculate the number of possible combinations:
𝑛 = 4 (number of straws)
𝑟 = 3 (number of straws needed for each combination)
𝐶(4, 3) = 4! / (3! (4 - 3)!)
= 4! / (3! x 1)
= 4 x 3 x 2 x 1 / (3 x 2 x 1 x 1)
= 4
So, there are 4 possible combinations of three straws that Joanna can make.
Now, let's see what those combinations are. We'll list them out:
Combination 1: Straws A, B, and C
Combination 2: Straws A, B, and D
Combination 3: Straws A, C, and D
Combination 4: Straws B, C, and D
These are the four different combinations of straws that Joanna can use to make triangles.
Now, let's make sure to remind Joanna not to start sword-fighting with these straws. Instead, she should enjoy creating different triangles and have fun exploring the world of geometry!
To determine how many different triangles Joanna can make using combinations of three straws, we need to consider the different combinations of straws and check if they satisfy the triangle inequality.
The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's consider all possible combinations of three straws:
1. Combination ABC:
- A = 7 inches
- B = 9 inches
- C = 3 inches
Checking the triangle inequality: A + B > C
7 + 9 = 16 > 3
The combination ABC forms a triangle.
2. Combination ABD:
- A = 7 inches
- B = 9 inches
- D = 10 inches
Checking the triangle inequality: A + B > D
7 + 9 = 16 > 10
The combination ABD forms a triangle.
3. Combination ACD:
- A = 7 inches
- C = 3 inches
- D = 10 inches
Checking the triangle inequality: A + C > D
7 + 3 = 10 > 10
The combination ACD does not form a triangle.
4. Combination BCD:
- B = 9 inches
- C = 3 inches
- D = 10 inches
Checking the triangle inequality: B + C > D
9 + 3 = 12 > 10
The combination BCD forms a triangle.
Therefore, Joanna can make three different triangles using combinations of three straws: ABC, ABD, and BCD.
To determine how many different triangles Joanna can make using the given straws, we need to consider the triangle inequality theorem. According to this theorem, for three side lengths to form a triangle:
- The sum of the lengths of any two sides must be greater than the length of the third side.
Using this criterion, we can go through all possible combinations of three straws and check if they satisfy this condition.
First, let's list all the combinations of three straws:
Combination 1: A, B, C
Combination 2: A, B, D
Combination 3: A, C, D
Combination 4: B, C, D
Now, let's check if these combinations satisfy the triangle inequality theorem.
Combination 1: A, B, C
A + B = 7 + 9 = 16 > C = 3 (Satisfied)
B + C = 9 + 3 = 12 > A = 7 (Satisfied)
A + C = 7 + 3 = 10 > B = 9 (Satisfied)
Combination 2: A, B, D
A + B = 7 + 9 = 16 > D = 10 (Satisfied)
B + D = 9 + 10 = 19 > A = 7 (Satisfied)
A + D = 7 + 10 = 17 > B = 9 (Satisfied)
Combination 3: A, C, D
A + C = 7 + 3 = 10 > D = 10 (Satisfied)
C + D = 3 + 10 = 13 > A = 7 (Satisfied)
A + D = 7 + 10 = 17 > C = 3 (Satisfied)
Combination 4: B, C, D
B + C = 9 + 3 = 12 > D = 10 (Satisfied)
C + D = 3 + 10 = 13 > B = 9 (Satisfied)
B + D = 9 + 10 = 19 > C = 3 (Satisfied)
From the above calculations, we can see that all four combinations satisfy the triangle inequality theorem. Therefore, Joanna can make a different triangle with each combination of straws.
In total, Joanna can make four different triangles using the given straws. The combinations she uses are:
1. A, B, C
2. A, B, D
3. A, C, D
4. B, C, D