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chord

 
Dictionary: chord2   (kôrd, kōrd) pronunciation
 
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n.
  1. A line segment that joins two points on a curve.
  2. A straight line connecting the leading and trailing edges of an airfoil.
  3. Anatomy. Variant of cord (sense 5).
  4. An emotional feeling or response: Her words struck a sympathetic chord in her audience.
  5. Archaic. The string of a musical instrument.

[Alteration of CORD.]


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Columbia Encyclopedia: chord
Home > Library > Miscellaneous > Columbia Encyclopedia
chord (kôrd) , in geometry, straight line segment both end points of which lie on the circumference of a circle or other curve; it is a segment of a secant. A chord passing through the center of a circle is a diameter. In the same circle or in equal circles, equal chords subtend equal arcs and equal central angles.

 
Word Tutor: chord
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pronunciation

IN BRIEF: A straight line drawn inside a circle and touching the circle at each end. Also: Three or more notes in music played together.

pronunciation Every action of our lives touches on some chord that will vibrate in eternity. — Edwin Hubble Chapin (1814-1880)/

Tutor's tip: A "chord" is musical notes played together for harmony, a "cord" is a material used to tie things, while "cored" is the past tense of core, which is to remove the center of a fruit.

 
Wikipedia: Chord (geometry)
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A chord of a curve is a geometric line segment whose endpoints both lie on the curve. A secant or a secant line is the line extension of a chord.

The red line BX is a chord.

Contents

Chords of a circle

Further information: Chord properties

Among properties of chords of a circle are the following:

  1. Chords are equidistant from the center if and only if their lengths are equal.
  2. A chord's perpendicular bisector passes through the centre.
  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.

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.

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. By taking one of the points to be zero, it can easily be related to the modern sine function:

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

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, not extant, 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 sin2θ + cos2θ = 1 \mbox{crd}^2 \theta + \mbox{crd}^2 (180^{\circ} - \theta) = 4
Half-angle \sin\frac{\theta}{2} = \pm\sqrt{\frac{1-\cos^2 \theta}{2}} \mbox{crd}\ \frac{\theta}{2} = \pm \sqrt{2-\mbox{crd}(180^{\circ} - \theta)}

The half-angle identity greatly expedites the creation of chord tables. Ancient chord tables typically used a large value for the radius of the circle, and reported the chords for this circle. It was then a simple matter of scaling to determine the necessary chord for any circle. According to G. J. Toomer, Hipparchus used a circle of radius 3438' (=3438/60=57.3). This value is extremely close to 180 / π (=57.29577951...). One advantage of this choice of radius was that he could very accurately approximate the chord of a small angle as the angle itself. In modern terms, it allowed a simple linear approximation:

\frac{3438}{60} \mbox{crd}\ \theta = 2 \frac{3438}{60} \sin \frac{\theta}{2} \approx 2 \frac{3438}{60} \frac{\pi}{180} \frac{\theta}{2} = \left(\frac{3438}{60} \frac{\pi}{180}\right) \theta \approx \theta.

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This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

 
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Copyrights:

Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2007. Published by Houghton Mifflin Company. All rights reserved.  Read more
Columbia Encyclopedia. The Columbia Electronic Encyclopedia, Sixth Edition Copyright © 2003, Columbia University Press. Licensed from Columbia University Press. All rights reserved. www.cc.columbia.edu/cu/cup/  Read more
Word Tutor. Copyright © 2004-present by eSpindle Learning, a 501(c) nonprofit organization. All rights reserved.
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Wikipedia. This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Chord (geometry)".  Read more

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