2005 AMC 8 Problems/Problem 21
Problem
How many distinct triangles can be drawn using three of the dots below as vertices?
Solution 1
The number of ways to choose three points to make a triangle is . However, two* of these are a straight line so we subtract to get .
- Note: We are assuming that there are no degenerate triangles in this problem, and that is why we subtract two.
Solution 2
Case 1: One vertex is on the top 3 points
Here, there is ways to choose the vertex on the top and ways the choose the on the bottom, so there is triangles.
Case 2: One vertex is on the bottom 3 points
By symmetry, there is triangles.
Otherwise, the triangle is degenerate.
Our final answer is ~Ddk001
Video solution
https://www.youtube.com/watch?v=XQS-KVW1O6M ~David
See Also
2005 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
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