Six Related Lines

Jon Ingram
These six lines are all related to each other. There are some questions below which ask you to think about various features of the lines (and particularly where the lines intersect with each other).
This applet shows six related lines. They are generated using three numbers. If you select the 'Show sliders' option, you can see the numbers being used, labelled , , and . 1) Can you explain how the three numbers are used to make the six lines? [ Hint: One of the lines is ] You can see the equations by selecting the 'Show equations' option. 2) Is there a reason why the six lines are coloured in the way they are? 3) Stop the animation, if you haven't already, and experiment with altering each of the numbers. Can you predict in advance what changing each number will do to the picture? 3) When are there less than six distinct lines? What numbers of lines are possible? In particular, is it possible to get exactly 2, or 4 lines? 4) Do you notice any symmetry relationships? Can you describe why the symmetry exists? 5) Do you notice anything interesting about where the lines intersect each other? If not, try selecting the 'Show extra lines' option. Does that help you notice anything? 5) What combinations of gradients are possible? In particular, what combinations of parallel lines are possible?