Chord (geometry)

A chord (from the Latin chorda, meaning "

(Latin for "arrow").

More generally, a chord is a line segment joining two points on any curve, for instance, on an ellipse. A chord that passes through a circle's center point is the circle's diameter.

In circles

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

In trigonometry

Chords were used extensively in the early development of

The chord function is defined geometrically as shown in the picture. The chord of an

${\displaystyle \operatorname {crd} \ \theta ={\sqrt {(1-\cos \theta )^{2}+\sin ^{2}\theta }}={\sqrt {2-2\cos \theta }}=2\sin \left({\frac {\theta }{2}}\right).}$ [3]

The last step uses the

. 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:^[4]

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

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
  - (${\displaystyle \operatorname {crd} \theta =2{\sqrt {\operatorname {haversin} \theta }}}$)
• 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

External links

Wikimedia Commons has media related to Chord (geometry).

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