Secant Line
A secant line, also simply called a secant, is a line passing through two points of a curve. As the two points are brought together (or, more precisely, as one is brought towards the other), the secant line tends to a tangent line.
The secant line connects two points (x,f(x)) and (a,f(a)) in the Cartesian plane on a curve described by a function y=f(x). It gives the average rate of change of f from x to a
| [画像: A(x)=(f(x)-f(a))/(x-a), ] |
(1)
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which is the slope of the line connecting the points (x,f(x)) and (a,f(a)). The limiting value
as the point a approaches x gives the instantaneous slope of the tangent line to f(x) at each point x, which is a quantity known as the derivative of f(x), denoted f^'(x) or df/dx.
The use of secant lines to iteratively find the root of a function is known as the secant method.
In abstract mathematics, the points connected by a secant line can be either real or complex conjugate imaginary.
In geometry, a secant line commonly refers to a line that intersects a circle at exactly two points (Rhoad et al. 1984, p. 429). There are a number of interesting theorems related to secant lines.
In the left figure above,
| theta=1/2(arcAC+arcBD), |
(3)
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while in the right figure,
| phi=1/2(arcRT-arcSQ), |
(4)
|
where arcAB denotes the angular measure of the arc AB (Jurgensen 1963, pp. 336-337).
See also
Arc, Average Rate of Change, Bitangent, Chord, Circle, Circle-Line Intersection, Secant Method, Tangent Line, Transversal LineExplore with Wolfram|Alpha
More things to try:
References
Jurgensen, R. C.; Donnelly, A. J.; and Dolciani, M. P. Th. 42 in Modern Geometry: Structure and Method. Boston, MA: Houghton-Mifflin, 1963.Rhoad, R.; Milauskas, G.; and Whipple, R. Geometry for Enjoyment and Challenge, rev. ed. Evanston, IL: McDougal, Littell & Company, 1984.Referenced on Wolfram|Alpha
Secant LineCite this as:
Weisstein, Eric W. "Secant Line." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SecantLine.html