The number of triangles with integer sides and with perimeter 15 is ………… .
The number of triangles with integer sides and with perimeter 15 is ………… .
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Solution and Explanation
Understanding the Triangle Inequality: For a triangle with sides \(a\), \(b\), and \(c\), the triangle inequality must hold:
- \(a + b > c\)
- \(a + c > b\)
- \(b + c > a\)
Finding the Combinations:
- Start with the largest possible side: Let's assume the largest side is 7. Then the remaining two sides must sum to 8. Possible combinations are (7, 7, 1), (7, 6, 2), (7, 5, 3), (7, 4, 4). All of these satisfy the triangle inequality.
- Next largest side is 6: Remaining two sides sum to 9. Possible combinations are (6, 6, 3), (6, 5, 4).
- Next largest side is 5: Remaining two sides sum to 10. The possible combination is (5, 5, 5).
Checking for Duplicates: We have listed all possible combinations without repeating any.
Counting the Triangles: We have found 7 unique combinations:
- (7, 7, 1)
- (7, 6, 2)
- (7, 5, 3)
- (7, 4, 4)
- (6, 6, 3)
- (6, 5, 4)
- (5, 5, 5)
Answer: The number of triangles with integer sides and perimeter 15 is 7.
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