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 [[Category:Geometry]]   [[Category:Geometry]] 
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−  Number of Chords formed by n points on a circle
 
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−  The formula for finding the number of chords is <math>\dfrac{n(n+1)}{2}  n</math> or <math>\dfrac{n(n1)}{2}</math>
 
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−  Method:
 
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−  Begin creating circles with an ascending number of points:
 
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−  1 point 0 chords
 
−  2 points 1 chord
 
−  3 points 3 chords
 
−  4 points 6 chords
 
−  5 points 10 chords
 
−  6 points 15 chords
 
−  7 points 21 chords
 
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−  On the right column, the numbers (from top to bottom) are ascending in a sequence known as the triangular numbers. This occurred every time you increase the number of points on a circle by 1. However, this shows that the 1st triangular number (1 chord) was the result of 2 points, the 2nd triangular number (3 chords) was the result of 3 points, the 3rd triangular number (6 chords) was the result of 4 points, etc.
 
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−  If the numbers on the left column are marking each triangular number listed as the first, second, third, etc.(1st point, 2nd point, 3rd point, etc.,), then something has been altered  the numbers on the left column have been moved backward! This means that when finding the number of chords created by "n" points on a line, you would have to subtract "n" from the "n"th triangular number! To find the "n"th triangular number, the formula "n(n+1)/2" is used.
 
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−  Subtracting "n" from this formula gives a new formula:
 
−  n(n+1)/2  n
 
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−  2(2+1)/2  2 = 6/2  2 = 32 = 1
 
−  3(3+1)/2  3 = 12/2  3 = 63 = 3
 
−  4(4+1)/2  4 = 20/2  4 = 104 = 6
 
−  5(5+1)/2  5 = 30/2  5 = 155 = 10
 
−  6(6+1)/2  6 = 42/2  6 = 216 = 15
 
−  7(7+1)/2  7 = 56/2  7 = 287 = 21
 
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−  In conclusion, if "n" points are placed on a circle, then the maximum number of chords that can connect any two points on that circle is:
 
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−  n(n+1)/2  n or n(n1)/2 (the simplified version)
 
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−  Note: In order to achieve the simplified version, use this process:
 
−  n(n+1)/2  n = (n(n+1)2n)/2 = (n^2n)/2 = (n1)n/2
 
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−  This method and formula was created by AoPS member Keshav Ramesh (user kr1234)
 
Latest revision as of 11:16, 18 February 2018
A chord of a circle is a line segment joining two points on .
The diameter of a circle is the longest chord of that circle. The diameter goes through the center of the circle.
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