I can only get 14, but I know there are more. What is the largest number of regions into which you can divide a circle with six lines? Is there a formula? How can you get this answer? I've tried drawing diagrams but I can't find a way to make sure they are correct. Then we can subtract the first of these two from the second: $$1=2a$$ so that \(a=\frac$$ Drawing six chordsĪ similar question was asked in 2001, and got two answers: Dividing a Circle using Six Lines One way to solve the system of equations is to first subtract each equation from the next one, leaving two equations in two unknowns: $$2=3a b\\3=5a b$$ You can then verify that if x = 4, y turns out to be 11 just as in the chart. Where x is the number of chords and y is the number of regions formed. So, substituting them into y = ax^2 bx c we get The solution to this system is a = 1/2, b = 1/2, and c = 1. Substitute them into the standard quadratic form and get We’re going to solve for the three coefficients of the formula. Substitute those numbers back into the standard equation and we'll have finished. We will then have three equations in three variables which we can solve. So, we can pick any three ordered pairs from the chart, substitute the values into the standard equation of a quadratic function (y = ax^2 bx c). For example, looking at the perfect squares, 1, 4, 9, 16, we see that the differences from one term to the next are 3, 5, 7, and so on, always increasing by 2. Here are the first few cases, showing how each new chord n crosses each of the previous n – 1 chords, cutting off n new regions (one from each of the n regions it passed through), thereby adding 1, then 2, then 3, then 4 regions:ĭoctor Jaffee has made a table of values, then used the fact that if the difference increases by the same amount at each step, the formula will have quadratic form. In the example above each increase is 1 more than the previous increase. You noticed that the number of regions increases by 2, then 3, then 4, etc., and that is one of the properties of quadratic functions that is, as the x number increases by 1, the y number increases by a constantly increasing amount. We can use that to construct a table of values that can be helpful: Your observation about how the number of regions increases with each additional chord is right on target. On the other hand, if any chord does not intersect all the others, we again have fewer than 7 regions:ĭoctor Jaffee replied, using an approach based on looking for a pattern in the numbers: Hi Dana, For example, if my three chords intersected at one point, we would lose a region, and have only six: We’ll be seeing different characterizations of the same problem, in which we are only asked for the maximum number of regions that will turn out to imply this idea that each chord intersects all the others. Here is a simple example, with 3 chords dividing the circle into 7 regions: I'm having trouble finding out the formula to solve the problem. This is how much I've completed on this problem: I know that the xth line splits x regions, increasing the number of sections by x. How many individual regions are in the circle? Each chord crosses every other chord but no three meet at the same point. We’ll see how several Math Doctors solved the problem, each from a sufficiently different perspective that it might communicate better to certain readers so if you don’t follow one, keep reading! Finding a quadratic formulaįor the first problem, we can start with this question from 1998: Regions, Chords, and CirclesĪ given circle has n chords. In each, what we count will be the regions into which the circle is cut. We’ve looked at how to count diagonals in a polygon this week and next, I want to consider two different problems (though they look similar at first) dealing with chords of a circle (which are practically the same thing as diagonals of a polygon).
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