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


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,


while in the right figure,


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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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 Line

Cite this as:

Weisstein, Eric W. "Secant Line." From MathWorld--A Wolfram Web Resource.

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