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Chord (geometry)

Geometric line segment whose endpoints both lie on the curve

Summary

Geometric line segment whose endpoints both lie on the curve

the line segment defined on a curve

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

A chord (from the Latin chorda, meaning "catgut or string") of a circle is a straight line segment whose endpoints both lie on a circular arc. If a chord were to be extended infinitely on both directions into a line, the object is a secant line. The perpendicular line passing through the chord's midpoint is called sagitta (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

Main article: Circle#Chord

Among properties of chords of a circle 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 a conic are collinear (midpoint theorem for conics).

{{anchor|crd|acrd}}In trigonometry

Chords were used extensively in the early development of trigonometry. The first known trigonometric table, compiled by Hipparchus in the 2nd century BC, is no longer extant but tabulated the value of the chord function for every degrees. In the 2nd century AD, Ptolemy compiled a more extensive table of chords in his book on astronomy, giving the value of the chord for angles ranging from to 180 degrees by increments of degree. Ptolemy used a circle of diameter 120, and gave chord lengths accurate to two sexagesimal (base sixty) digits after the integer part.

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 {{math|0

: \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	\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 ^2-4 a^2}
Angle (θ)	c=2 r \sin \left(\frac{\theta }{2}\right)	c=\frac{D}{2} \operatorname{crd} \theta

The inverse function exists as well:

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

References

Gibson, C. G.. (2003). "Elementary Euclidean Geometry: An Introduction". Cambridge University Press.
Maor, Eli. (1998). "Trigonometric Delights". Princeton University Press.
Weisstein, Eric W. "[https://mathworld.wolfram.com/CircularSegment.html Circular Segment]". From MathWorld--A Wolfram Web Resource.
Simpson, David G.. (2001-11-08). "AUXTRIG". NASA Goddard Space Flight Center.

Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

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