World Library  
Flag as Inappropriate
Email this Article

Chord (geometry)

Article Id: WHEBN0000297947
Reproduction Date:

Title: Chord (geometry)  
Author: World Heritage Encyclopedia
Language: English
Subject: Line segment, History of trigonometry, Trigonometry, Trigonometric functions, Equichordal point
Collection:
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Chord (geometry)

A chord of a circle is a geometric line segment whose endpoints both lie on the circle. A secant line, or just secant, is the line extension of a chord. More generally, a chord is a line segment joining two points on any curve, for instance an ellipse. A chord that passes through a circle's center point is the circle's diameter.

The red segment BX is a chord
(as is the diameter segment AB).

Chords of a circle

Among properties of chords of a circle are the following:

  1. Chords are equidistant from the center only if their lengths are equal.
  2. A chord that passes through the center of a circle is called a diameter, and is the longest chord.
  3. If the line extensions (secant lines) of chords AB and CD intersect at a point P, then their lengths satisfy AP·PB = CP·PD (power of a point theorem).

The area that a circular chord "cuts off" is called a circular segment.

Chord is from the Latin chorda meaning bowstring.

Chords of an ellipse

The midpoints of a set of parallel chords of an ellipse are collinear.[1]:p.147

Chords in trigonometry

Chords were used extensively in the early development of trigonometry. The first known trigonometric table, compiled by Hipparchus, tabulated the value of the chord function for every 7.5 degrees. In the second century AD, Ptolemy of Alexandria compiled a more extensive table of chords in his book on astronomy, giving the value of the chord for angles ranging from 1/2 degree to 180 degrees by increments of half a degree. The circle was of diameter 120, and the chord lengths are accurate to two base-60 digits after the integer part.

The chord function is defined geometrically as in the picture to the left. The chord of an angle is the length of the chord between two points on a unit circle separated by that angle. The chord function can be related to the modern sine function, by taking one of the points to be (1,0), and the other point to be (cos \theta, sin \theta), and then using the Pythagorean theorem to calculate the chord length:

\mathrm{crd}\ \theta = \sqrt{(1-\cos \theta)^2+\sin^2 \theta} = \sqrt{2-2\cos \theta} =2 \sin \left(\frac{\theta }{2}\right).

The last step uses the half-angle formula. Much as modern trigonometry is built on the sine function, ancient trigonometry was built on the chord function. Hipparchus is purported to have written a twelve volume work on chords, all now lost, so presumably a great deal was known about them. The chord function satisfies many identities analogous to well-known modern ones:

Name Sine-based Chord-based
Pythagorean \sin^2 \theta + \cos^2 \theta = 1 \, \mathrm{crd}^2 \theta + \mathrm{crd}^2 (180^\circ - \theta) = 4 \,
Half-angle \sin\frac{\theta}{2} = \pm\sqrt{\frac{1-\cos \theta}{2}} \, \mathrm{crd}\ \frac{\theta}{2} = \pm \sqrt{2-\mathrm{crd}(180^\circ - \theta)} \,
Apothem (a) c=2 \sqrt{r^2- a^2} c=\sqrt{D ^2-4 a^2}
Angle (θ) c=2 r \sin \left(\frac{\theta }{2}\right) c=D \sin \left(\frac{\theta }{2}\right)

See also

References

  1. ^ Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979.

External links

  • History of Trigonometry Outline
  • Trigonometric functions, focusing on history
  • Chord (of a circle) With interactive animation
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 



Copyright © World Library Foundation. All rights reserved. eBooks from World eBook Library are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.