To determine how many different triangles Joanna can make, we need to consider the combinations of three straws she can use.
Using the formula for combinations, we can calculate the total number of combinations of 4 straws taken 3 at a time:
C(4, 3) = 4! / (3!(4-3)!) = 4
These are the possible combinations of straws Joanna can use to make triangles:
1. A, B, C
2. A, B, D
3. A, C, D
4. B, C, D
Now, we need to check if each combination of straws can form a triangle based on the triangle inequality theorem, which 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 check each combination:
1. A, B, C:
- A + B = 7 + 9 = 16 > 3 (C)
- A + C = 7 + 3 = 10 > 9 (B)
- B + C = 9 + 3 = 12 > 7 (A)
Triangle can be formed with sides A, B, and C.
2. A, B, D:
- A + B = 7 + 9 = 16 > 10 (D)
- A + D = 7 + 10 = 17 > 9 (B)
- B + D = 9 + 10 = 19 > 7 (A)
Triangle can be formed with sides A, B, and D.
3. A, C, D:
- A + C = 7 + 3 = 10 > 10 (D)
- A + D = 7 + 10 = 17 > 3 (C)
- C + D = 3 + 10 = 13 > 7 (A)
Triangle can be formed with sides A, C, and D.
4. B, C, D:
- B + C = 9 + 3 = 12 > 10 (D)
- B + D = 9 + 10 = 19 > 3 (C)
- C + D = 3 + 10 = 13 > 9 (B)
Triangle can be formed with sides B, C, and D.
Therefore, Joanna can make 4 different triangles using combinations of the straws.
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.
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