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Secant Line


TangentSecantLine

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

 A(x)=(f(x)-f(a))/(x-a),
(1)

which is the slope of the line connecting the points (x,f(x)) and (a,f(a)). The limiting value

 f^'(x)=lim_(a->x)(f(x)-f(a))/(x-a)
(2)

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.

SecantTheorems

In the left figure above,

 theta=1/2(arcAC+arcBD),
(3)

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 Line

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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.

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Secant Line

Cite this as:

Weisstein, Eric W. "Secant Line." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SecantLine.html

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