# Difference between revisions of "Chord"

I like pie (talk | contribs) m |
|||

Line 21: | Line 21: | ||

{{stub}} | {{stub}} | ||

[[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 |

## Revision as of 15:38, 27 September 2016

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.

*This article is a stub. Help us out by expanding it.*

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