Grades 9-10 Video Solutions 2024-2024_9-10

Video Transcription

There are n distinct lines on the plane labeled L1 through Ln. The line L1 intersects exactly 5 other lines, the line L2 intersects exactly 9 other lines,  the line L3 intersects exactly 11 other lines. What is the minimum possible value of n?  First note that because we have a line intersecting 11 other lines, the smallest possible number anyways is 12. Now let's consider families of parallel lines.  If we have a family of 7 parallel lines, a family of 3, and then a family of 2, each of the families not being parallel to each other and intersecting each other, then we  have an arrangement of lines that meets this requirement. Note that any of the lines in the first family can be L1 and it will intersect 5 other lines,  the family of 3 and the family of 2. Any of the lines in the family of 3 can be L2 because it will intersect the family of 7 and the family of 2.  Any of the last two lines in that family can be L3 because it will intersect 11 other lines. Therefore our final answer is 12.

Video Summary

The problem involves determining the smallest number of distinct lines on a plane, where specific lines intersect a given number of other lines. L1 intersects 5 lines, L2 intersects 9 lines, and L3 intersects 11 lines. Recognizing that a line intersecting 11 others implies a minimum of 12 lines, the proposed solution includes three families: 7 parallel lines, 3 parallel lines, and 2 parallel lines, with each family intersecting the others. This configuration allows L1 to intersect 5 lines, L2 to intersect 9, and L3 to intersect 11, satisfying all conditions. Thus, the minimum value of n is 12.

Keywords

distinct lines

intersection

minimum lines

parallel lines

plane geometry