Difference between revisions of "Chord"

 
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[[Category:Geometry]]
 
[[Category:Geometry]]
 
 
Number of Chords formed by n points on a circle
 
 
The formula for finding the number of chords is n(n+1)/2 - n or n(n-1)/2
 
 
 
Method:
 
 
Begin creating circles with an ascending number of points:
 
 
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
 
 
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.
 
 
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.
 
 
Subtracting "n" from this formula gives a new formula:
 
n(n+1)/2 - n
 
 
2(2+1)/2 - 2      =      6/2 - 2      =      3-2    =    1
 
3(3+1)/2 - 3      =      12/2 - 3    =      6-3    =    3
 
4(4+1)/2 - 4      =      20/2 - 4    =      10-4  =    6
 
5(5+1)/2 - 5      =      30/2 - 5    =      15-5  =    10
 
6(6+1)/2 - 6      =      42/2 - 6    =      21-6  =    15
 
7(7+1)/2 - 7      =      56/2 - 7    =      28-7  =    21
 
 
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:
 
 
n(n+1)/2 - n  or  n(n-1)/2 (the simplified version)
 
 
Note: In order to achieve the simplified version, use this process:
 
n(n+1)/2 - n = (n(n+1)-2n)/2 = (n^2-n)/2 = (n-1)n/2
 
 
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 $O$ is a line segment joining two points on $O$.

[asy]size(100); pair O=origin,A=dir(135),B=dir(30); D(unitcircle); D(A--B); MP("O",D(O),S); MP("A",D(A),W); MP("B",D(B),E);[/asy]

The diameter of a circle is the longest chord of that circle. The diameter goes through the center of the circle.

[asy]size(120); pair O=origin,A=dir(170),B=dir(-10); D(unitcircle); D(A--B); MP("O",D(O),N); MP("A",D(A),W); MP("B",D(B),E);[/asy]

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