Problem No. 22. Lines on a plane

Let's draw lines!

Is it possible that 7 lines on a plane have 30 intersection points in total? Can they have twenty intersection points?

Student tip

Find the maximum number of intersection points. Start with two lines, add one at a time, until you reveal the pattern.

Solution

\(n\) lines have maximum of \(n(n-1)/2\) intersection points in total, which means that 7 lines have maximum of 21 intersection points in total, and it is not possible to have 30 intersection points. They can have 20 intersection points, if two lines are parallel, and each two lines (except the mentioned pair) intersect at different points.