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Question

AMC 8 practice problem number 25. Please explain how to do the problem.

Three points are chosen on each side of a square, aside from the vertices of the square. Lines are drawn connecting every pair of the 12 points that are on different sides of the square. If no three lines concur at the same point on the square, at how many points do two of the lines intersect inside the square?

Answer 1

Let's break down the problem step by step.

1. Choosing Points:
2. There are 4 sides of the square.
3. On each side, 3 points are chosen, excluding the vertices of the square.
4. So, there are a total of 4 sides * 3 points/side = 12 points chosen.
5. Connecting Lines:
6. Lines are drawn between every pair of the 12 points that are on different sides of the square.
7. This means each point on a side is connected to every point on the other three sides.
8. Finding Intersections:
9. We are interested in finding how many points of intersection occur inside the square.
10. To find the number of intersection points, we need to count the number of intersections for each pair of lines.
11. Counting Intersections:
12. Consider any two lines. Since no three lines concur at the same point on the square, the lines must intersect inside the square.
13. So, for each pair of lines, there is exactly one intersection point inside the square.
14. Counting Pairs:
15. The number of ways to choose 2 lines out of 12 is given by the combination formula: C(n, k) = n! / (k!(n-k)!), where n is the total number of lines (12) and k is the number of lines to choose (2).
16. C(12, 2) = 12! / (2!(12-2)!) = 66.
17. Conclusion:
18. There are 66 intersection points inside the square.

In summary, you can find the number of intersection points by considering each pair of lines and realizing that each pair results in one intersection point inside the square.