Given five segments of length 2,3,5,8, and 13, what is the number of distinct triangles that can be formed using any three of the segments?
To determine the number of distinct triangles that can be formed using any three segments, we need to consider the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining side.
Let's go step by step to solve the problem:
1. Sort the given segments in ascending order: 2, 3, 5, 8, 13.
2. Start by taking the smallest two segments and check if their sum is greater than the remaining segment:
- The segments 2 and 3 cannot form a triangle because 2 + 3 = 5, which is equal to the length of the remaining longest segment, 5.
- Move on to the next pair.
3. Take the pair (2, 5) and check if their sum is greater than the length of the remaining segment:
- The segments 2 and 5 can form a triangle because 2 + 5 = 7, which is greater than the length of the remaining segment, 3.
- Thus, we have found our first triangle.
4. Continue this process by considering the next pair (2, 8):
- The segments 2 and 8 can form a triangle because 2 + 8 = 10, which is greater than the length of the remaining segment, 3.
- We have found another triangle.
5. For the next pair (3, 5):
- The segments 3 and 5 can form a triangle because 3 + 5 = 8, which is greater than the length of the remaining segment, 2.
- We have found another triangle.
6. For the pair (3, 8):
- The segments 3 and 8 can form a triangle because 3 + 8 = 11, which is greater than the length of the remaining segment, 2.
- We have found another triangle.
7. Lastly, for the pair (5, 8):
- The segments 5 and 8 can form a triangle because 5 + 8 = 13, which is equal to the length of the remaining segment, 13.
- This is a degenerate triangle, where all three segments are collinear.
In total, we have found four triangles that can be formed using any three segments from the given set: (2, 5, 8), (2, 3, 5), (2, 3, 8), and (3, 5, 8).