Chord (geometry)

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

A chord of a circle is a straight line segment whose endpoints both lie on a circular arc. The infinite line extension of a chord is a secant line, or just secant. 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 word chord is from the Latin chorda meaning bowstring.

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

In circles

Main article: Circle § Chord

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. Equal chords are subtended by equal angles from the center of the circle.
3. A chord that passes through the center of a circle is called a diameter and is the longest chord of that specific circle.
4. 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 a conic are collinear (midpoint theorem for conics).^[1]

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+1/2 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 to 180 degrees by increments of 1/2 degree. The circle was of diameter 120, and the chord lengths are accurate to two base-60 digits after the integer part.^[2]

The chord function is defined geometrically as shown in the picture. The chord of an angle is the length of the chord between two points on a unit circle separated by that central angle. The angle θ is taken in the positive sense and must lie in the interval 0 < θ ≤ π (radian measure). 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 θ, sin θ), and then using the Pythagorean theorem to calculate the chord length:^[2]

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

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. In the table below (where c is the chord length, and D 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	${\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1\,}$	${\displaystyle \operatorname {crd} ^{2}\theta +\operatorname {crd} ^{2}(\pi -\theta )=4\,}$
Half-angle	${\displaystyle \sin {\frac {\theta }{2}}=\pm {\sqrt {\frac {1-\cos \theta }{2}}}\,}$	${\displaystyle \operatorname {crd} \ {\frac {\theta }{2}}={\sqrt {2-\operatorname {crd} (\pi -\theta )}}\,}$
Apothem (a)	${\displaystyle c=2{\sqrt {r^{2}-a^{2}}}}$	${\displaystyle c={\sqrt {D^{2}-4a^{2}}}}$
Angle (θ)	${\displaystyle c=2r\sin \left({\frac {\theta }{2}}\right)}$	${\displaystyle c={\frac {D}{2}}\operatorname {crd} \ \theta }$

The inverse function exists as well:^[3]

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

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
• Zindler curve (closed and simple curve in which all chords that divide the arc length into halves have the same length)

References

1. Chakerian, G. D. (1979). "7". In Honsberger, R. (ed.). A Distorted View of Geometry. Mathematical Plums. Washington, DC, USA: Mathematical Association of America. p. 147.
2. Maor, Eli (1998), Trigonometric Delights, Princeton University Press, pp. 25–27, ISBN 978-0-691-15820-4
3. Simpson, David G. (2001-11-08). "AUXTRIG" (FORTRAN-90 source code). Greenbelt, Maryland, USA: NASA Goddard Space Flight Center. Retrieved 2015-10-26.

Further reading

Hawking, S.W., ed. (2002). On the Shoulders of Giants: The Great Works of Physics and Astronomy. Philadelphia, PA: Running Press. ISBN 0-7624-1698-X. LCCN 2002100441. Retrieved 2017-07-31.{{cite book}}: CS1 maint: url-status (link)

Stávek, Jiří (2017-03-10) [2017-02-26]. "On the hidden beauty of trigonometric functions". Applied Physics Research. 9 (2): 57–64. doi:10.5539/apr.v9n2p57. ISSN 1916-9639. ISSN 1916-9647. from the original on 2017-07-31. Retrieved 2021-10-21 – via Canadian Center of Science and Education.

External links

 

Wikimedia Commons has media related to Chord (geometry).

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