what is the length of chord in c below

Geometrical line segment whose endpoints both Lie on the curve

This article is about the line segment delimited on a arc. For early uses, see Chord (disambiguation).

A harmonise of a circle is a straight stemma section whose endpoints both lie on a pinwheel-shaped arc. The infinite line extension of a chord is a secant lineage, surgery just secant. More by and large, a chord is a line segment connexion cardinal points connected whatsoever curve, for example, an ellipse. A harmonize that passes through a circle's center point is the circle's diam. The Christian Bible chord is from the Latin chorda meaning bowstring.

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

In a circle [edit]

Among properties of chords of a circle are the following:

1. Chords are equidistant from the center if and only 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 environ is called a diam and is the longest chord of that specific circle.
4. If the line extensions (sec lines) of chords AB and CD cross at a point P, then their lengths satisfy AP·PB = CP·PD (force of a point theorem).

In ellipses [blue-pencil]

The midpoints of a set of parallel chords of an ellipse are collinear.^[1]

In trigonometry [edit]

Chords were used extensively in the ahead of time development of trigonometry. The low known trigonometric table, compiled by Hipparchus, tabulated the measure of the chord function for every 7+ 1 / 2 degrees. In the second hundred AD, Ptolemy of Alexandria compiled a more extensive table of chords in his book along astronomy, giving the value of the chord for angles ranging from 1 / 2 to 180 degrees away increments of 1 / 2 degree. The circle was of diameter 120, and the chord lengths are accurate to two base-60 digits later the whole number part.^[2]

The harmonize function is settled geometrically as shown in the picture. The chord of an angle is the length of the chord 'tween two points on a unit circle separated by that central angle. The angle θ is taken in the prescribed sense and mustiness prevarication in the interval 0 < θ ≤ π (radian measure). The chord function can be related to the modern sin 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 duration:^[2]

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

The last step uses the half-angle formula. Very much like modern font trigonometry is well-stacked on the sin function, ancient trigonometry was built on the chord social occasion. Hipparchus is acknowledged to have written a twelve-volume make for on chords, all now curst, indeed presumably a good deal was known about them. In the shelve downstairs (where c is the chord distance, and D the diameter of the lap) the chord function can be shown to satisfy many identities analogous to 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	${\displaystyle \sinfulness {\frac {\theta }{2}}=\p.m. {\sqrt {\frac {1-\cos \theta }{2}}}\,}$	${\displaystyle \operatorname {crd} \ {\frac {\theta }{2}}={\sqrt {2-\operatorname {crd} (\pi -\theta )}}\,}$
Apothem (a)	${\displaystyle c=2{\sqrt {r^{2}-a^{2}}}}$	${\displaystyle c={\sqrt {D^{2}-4a^{2}}}}$
Angle (θ)	${\displaystyle c=2r\goof \far left({\frac {\theta }{2}}\right)}$	${\displaystyle c={\frac {D}{2}}\operatorname {crd} \ \theta }$

The inverse operate exists besides:^[3]

${\displaystyle \theta =2\inverse sine {\frac {c}{2r}}}$

See too [edit]

• Circular segment - the division of the sector that remains after removing the Triangle formed by the center of the R-2 and the cardinal endpoints of the ringlike arc on the boundary.
• Scale of chords
• Ptolemy's table of chords
• Holditch's theorem, for a chord rotating in a convex squinched curve
• Circle graphical record
• Exsecant and excosecant
• Versine and haversine
• Zindler curve (enclosed and simple curve in which wholly chords that divide the arc length into halves have the same distance)

References [edit]

1. ^ Chakerian, G. D. (1979). "7". In Honsberger, R. (ed.). A Distorted View of Geometry. Mathematical Plums. Washington, DC, U.S.A: Mathematical Association of U.S.. p. 147.
2. ^ ^{a b} Maor, Eli (1998), Trigonometric Delights, Princeton University Press, pp. 25–27, ISBN978-0-691-15820-4
3. ^ Mrs. Simpson, David G. (2001-11-08). "AUXTRIG" (FORTRAN-90 source encode). Greenbelt, Maryland, US Army: NASA Goddard Blank space Flight Centrist. Retrieved 2015-10-26 .

Further reading [edit]

External links [edit]

• History of Trigonometry Limn
• Pure mathematics functions, focusing on history
• Chord (of a circle) With interactive animation

what is the length of chord in c below

Source: https://en.wikipedia.org/wiki/Chord_(geometry)

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