To determine how many different triangles Joanna can make using her straws, we need to apply the triangle inequality theorem. The triangle inequality states that for any three lengths \( a \), \( b \), and \( c \) to form a triangle, the following conditions must be true:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)
Let's list the lengths of the straws:
- Straw A: 7 inches
- Straw B: 9 inches
- Straw C: 3 inches
- Straw D: 10 inches
Now we can examine the combinations of three straws from the four available straws (A, B, C, D).
The combinations of three straws are:
- A, B, C
- A, B, D
- A, C, D
- B, C, D
Now we'll check each combination to see if it can form a triangle.
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Combination A, B, C (7, 9, 3): \[ 7 + 9 = 16 > 3 \quad (valid) \ 7 + 3 = 10 \not> 9 \quad (not valid) \ 9 + 3 = 12 > 7 \quad (valid) \] This combination cannot form a triangle.
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Combination A, B, D (7, 9, 10): \[ 7 + 9 = 16 > 10 \quad (valid) \ 7 + 10 = 17 > 9 \quad (valid) \ 9 + 10 = 19 > 7 \quad (valid) \] This combination can form a triangle.
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Combination A, C, D (7, 3, 10): \[ 7 + 3 = 10 \not> 10 \quad (not valid) \ 7 + 10 = 17 > 3 \quad (valid) \ 3 + 10 = 13 > 7 \quad (valid) \] This combination cannot form a triangle.
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Combination B, C, D (9, 3, 10): \[ 9 + 3 = 12 > 10 \quad (valid) \ 9 + 10 = 19 > 3 \quad (valid) \ 3 + 10 = 13 > 9 \quad (valid) \] This combination can form a triangle.
Based on the analysis above:
- The combinations that can form a triangle are A, B, D (7, 9, 10) and B, C, D (9, 3, 10).
- The total number of different triangles Joanna can make is 2.
Summary:
Number of different triangles that can be formed: 2
Combinations of straws used:
- A, B, D (7, 9, 10)
- B, C, D (9, 3, 10)
1 answer