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The starting point I have been given for this investigation is three lines which intersect each other, as shown below. R I L The lines, which we will refer to as L, are of infinitive length, though I will only draw them long enough so that we can see the diagrams clearly. The points that the lines cross each other (intersections) I will refer to as I. The areas that are bounded by the lines, I will call regions and refer to them as R. To help me find a relationship between I, L and R, I will have to set a number of boundaries for my lines, if I did not do this then I would have an endless investigation. In this investigation we wish to investigate the maximum number of intersections (I) and regions (R) that a certain number of lines (L) can make. To do this I will start the investigations with simple diagrams and then build them up to complicated sketches, I will then put the results of the findings of the sketches in tables. If I find it necessary I will also draw graphs to show my results clearer, when I have done this, using the tables I will look for a relationship between the no. of lines, intersections and regions. I am now going to set boundaries for the lines so as my investigation will have a limit and so I will be able to find a pattern between the lines, intersections and regions. 1) The lines must be straight. e.g. 1 2 3 4 5 6 If the lines weren’t straight, and they were curved, as above, I could have 1 line with I=5 and R=7. If this rule wasn’t used, then I couldn’t find any relationship because 1 line could intersect itself numerous times. 2) Lines must be of a standard length. e.g. As you can see, it would be very difficult to find a pattern if the lines were not of similar length. To cancel this out I will make all the diagrams of similar length, so as I don’t have diagrams as above. 3) Lines must not be parallel. I will now do a mini investigation to see what happens if the lines are parallel. 1 2 L=1 1 2 3 L=2 R=2 R=3 I=0 I=0 1 2 3 4 L=3 R=4 I=0 I will now put these results into a table, so I can understand them easier L I R 1 0 2 2 0 3 3 0 4 From looking at the table I can now say, that for parallel lines, there is no intersections and the no. of regions equals the no. of lines plus 1. I can now say that for all parallel lines I=0 and R=L+1. I can now predict that for 5 lines, there will be no intersections and 6 regions. I will now sketch this out to see if I am correct. L=5 1 2 3 4 5 6 R=6 I=0 I can now see from this diagram that my prediction is correct, L=5 R=6 and I=0.
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