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.(4 points)
To form a triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. Let's check all possible combinations of three straws to see which ones satisfy this condition.
For combination A, B, and C:
7 + 9 > 3 (True)
7 + 3 > 9 (False)
9 + 3 > 7 (True)
For combination A, B, and D:
7 + 9 > 10 (True)
7 + 10 > 9 (True)
9 + 10 > 7 (True)
For combination A, C, and D:
7 + 3 > 10 (False)
7 + 10 > 3 (True)
3 + 10 > 7 (True)
For combination B, C, and D:
9 + 3 > 10 (True)
9 + 10 > 3 (True)
3 + 10 > 9 (True)
The combinations that satisfy the condition are A, B, and D; A, C, and D; and B, C, and D. Therefore, Joanna can make three different triangles using these combinations of straws.
To determine how many different triangles Joanna can make using the four straws, we need to consider the combinations of three straws.
Using the formula for combinations, we can calculate the total number of combinations:
C(n, r) = n! / r!(n-r)!
Here, n is the total number of items (4 straws) and r is the number of items chosen (3 straws).
C(4, 3) = 4! / 3!(4-3)!
= 4! / 3!1!
= 4
So, Joanna can make 4 different triangles using the 4 straws.
To determine the combinations of straws for each triangle, we can write them out step by step:
1. First, choose A, B, and C: (A, B, C)
2. Next, choose A, B, and D: (A, B, D)
3. Then, choose A, C, and D: (A, C, D)
4. Finally, choose B, C, and D: (B, C, D)
So, the combinations of straws Joanna can use for the different triangles are:
1. (A, B, C)
2. (A, B, D)
3. (A, C, D)
4. (B, C, D)