Grades 11-12 Video Solutions 2013-Levels 11&12 Video Solutions 2013 problem26

Video Transcription

Question number 26. There are some straight lines drawn on the plane.  Line A intersects exactly three other lines and line B intersects exactly four other lines.  Line C intersects exactly N other lines but not three or four.  Determine the number of lines drawn on the plane.  So let's draw our first line, line A, and let's observe that since A and B have a different number of intersections, they cannot be parallel lines.  So let me draw in B here.  And now more lines to achieve the correct number of intersections.  So line A needs exactly three intersections. One of them is the line B and I'll draw in two more lines  parallel to B as not to create intersections with B.  And B has four intersecting lines so one of them is A and then I'll draw in three more lines parallel to A  so that I don't mess up my intersections with line A  that I have already drawn in. And now we have seven lines and one of the black lines could possibly be line C.  So let us study these.  If one of these lines here parallel to B is line C,  then they already cross line A and they will cross the three lines parallel to A  so they will have four intersections and so that cannot be line C. Now if one of these is line C,  then they already cross B and they will cross the two lines parallel to B with three intersections, neither of these can be line C as well.  So we cannot add another line because then that line will intersect either A or B  and we can't have more intersections on those lines  so we must somehow remove one intersection from A and B and then add in one more line that intersects both.  So let us do that. I can erase here one of the lines intersecting B  and I'll erase one of the lines intersecting A and then add in a line that intersects both A and B and that line I will call  line C and then check that it has the correct properties.  So here is my candidate line C and let's count the intersections.  We have one intersection with A, one intersection with B  and then it will intersect the three black lines  so it has a total of five intersections which is permitted. Now line A still has three intersections and line B still crosses four other lines. So we have achieved what is required.  Of us and we see that the number of lines here is exactly six.

Video Summary

The problem involves determining the number of lines on a plane where lines A, B, and C intersect specific numbers of other lines. Line A intersects exactly three other lines, line B intersects four, and line C intersects a number different from three or four. Initially, seven lines are drawn, but adjustments are made to meet the conditions by removing existing intersections and adding a new line for line C. The solution finds that with careful planning and recognition of intersection constraints, the correct number of lines on the plane is six.

Keywords

plane

lines

intersections

geometry

constraints