|
|
(3 intermediate revisions by 2 users not shown) |
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
| |
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.
This article is a stub. Help us out by expanding it.