Secant line

Encyclopedia

A

is a line that (locally) intersects two point

s on the curve. The word

It can be used to approximate the tangent

to a curve

, at some point

s,

as

, or direction, is that of the tangent. In calculus, this idea is the basis of the geometric definition of the derivative

.

A chord

is the portion of a secant that lies within the curve.

**secant line**of a curveCurve

In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

is a line that (locally) intersects two point

Point (geometry)

In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

s on the curve. The word

*secant*comes from the LatinLatin

Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...

*secare*,*to cut*.It can be used to approximate the tangent

Tangent

In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

to a curve

Curve

In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

, at some point

*P*. If the secant to a curve is defined by two pointPoint (geometry)

In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

s,

*P*and*Q*, with*P*fixed and*Q*variable, as*Q*approaches*P*along the curve, the direction of the secant approaches that of the tangent at*P*, (assuming that the first-derivative of the curve is continuous at point*P*so that there is only one tangent). As a consequence, one could say that the limitLimit (mathematics)

In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...

as

*Q*approaches*P*of the secant's slopeSlope

In mathematics, the slope or gradient of a line describes its steepness, incline, or grade. A higher slope value indicates a steeper incline....

, or direction, is that of the tangent. In calculus, this idea is the basis of the geometric definition of the derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

.

A chord

Chord (geometry)

A chord of a circle is a geometric line segment whose endpoints both lie on the circumference of the circle.A secant or a secant line is the line extension of a chord. More generally, a chord is a line segment joining two points on any curve, such as but not limited to an ellipse...

is the portion of a secant that lies within the curve.

## See also

- ChordChord (geometry)A chord of a circle is a geometric line segment whose endpoints both lie on the circumference of the circle.A secant or a secant line is the line extension of a chord. More generally, a chord is a line segment joining two points on any curve, such as but not limited to an ellipse...
- RadiusRadiusIn classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...
- DiameterDiameterIn geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle...
- Tangent