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Chord (geometry)

From Wikipedia, the free encyclopedia

Geometric line segment whose endpoints both lie on the curve

This article is about the line segment defined on a curve. For other uses, seeChord (disambiguation).

Common lines and line segments on a circle, including a chord in blue

Achord (from the Latinchorda, meaning "catgut or string") of acircle is astraight line segment whose endpoints both lie on acircular arc. If a chord were to be extendedinfinitely on both directions into aline, the object is asecant line. The perpendicular line passing through the chord'smidpoint is calledsagitta (Latin for "arrow").

More generally, a chord is a line segment joining two points on anycurve, for instance, on anellipse. A chord that passes through a circle's center point is the circle'sdiameter.

In circles

Main article:Circle § Chord

Among properties of chords of acircle are the following:

Chords are equidistant from the center if and only if their lengths are equal.
Equal chords are subtended by equal angles from the center of the circle.
A chord that passes through the center of a circle is called a diameter and is the longest chord of that specific circle.
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).

In conics

The midpoints of a set of parallel chords of aconic arecollinear (midpoint theorem for conics).^[1]

In trigonometry

Chords were used extensively in the early development oftrigonometry. The first known trigonometric table, compiled byHipparchus in the 2nd century BC, is no longer extant buttabulated the value of the chord function for every⁠7+1/2⁠degrees. In the 2nd century AD,Ptolemy compiled a more extensive table of chords inhis book on astronomy, giving the value of the chord for angles ranging from⁠1/2⁠ to 180 degrees by increments of⁠1/2⁠ degree. Ptolemy used a circle of diameter 120, and gave chord lengths accurate to twosexagesimal (base sixty) digits after the integer part.^[2]

The chord function is defined geometrically as shown in the picture. The chord of anangle is thelength of the chord between two points on a unit circle separated by thatcentral angle. The angleθ is taken in the positive sense and must lie in the interval0 <θ ≤π (radian measure). The chord function can be related to the modernsine function, by taking one of the points to be (1,0), and the other point to be (cosθ, sinθ), and then using thePythagorean theorem to calculate the chord length:^[2]

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

^[3]

The last step uses thehalf-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. In the table below (wherec is the chord length, andD the diameter of the circle) the chord function can be shown to satisfy many identities analogous to well-known modern ones:

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

The inverse function exists as well:^[4]

\theta =2\arcsin {\frac {c}{2r}}

See also

Circular segment – the part of the sector that remains after removing the triangle formed by the center of the circle and the two endpoints of the circular arc on the boundary.
Scale of chords
Ptolemy's table of chords
Holditch's theorem, for a chord rotating in a convex closed curve
Circle graph
Exsecant and excosecant
Versine and haversine – ( $\operatorname {crd} \theta =2{\sqrt {\operatorname {haversin} \theta }}$ )
Zindler curve (closed and simple curve in which all chords that divide the arc length into halves have the same length)
Bertrand paradox (probability) average length of chord paradox

References

^Gibson, C. G. (2003)."7.1 Midpoint Loci".Elementary Euclidean Geometry: An Introduction. Cambridge University Press. pp. 65–68.ISBN 9780521834483.
^^a ^bMaor, Eli (1998).Trigonometric Delights. Princeton University Press. pp. 25–27.ISBN 978-0-691-15820-4.
^Weisstein, Eric W. "Circular Segment". From MathWorld--A Wolfram Web Resource.
^Simpson, David G. (2001-11-08)."AUXTRIG" (FORTRAN-90 source code). Greenbelt, Maryland, US: NASA Goddard Space Flight Center. Retrieved2015-10-26.

External links

Wikimedia Commons has media related toChord (geometry).

History of Trigonometry Outline
Trigonometric functions Archived 2017-03-10 at theWayback Machine, focusing on history
Chord (of a circle) With interactive animation

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InElements	Angle bisector theorem Exterior angle theorem Euclidean algorithm Euclid's theorem Geometric mean theorem Hinge theorem Inscribed angle theorem Intercept theorem Intersecting chords theorem Intersecting secants theorem Law of cosines Pons asinorum Pythagorean theorem Tangent-secant theorem Thales's theorem Theorem of the gnomon

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Platonic Academy

School of Chios

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