# Exploring the Geometry of Circles: Tangents, Secants, and Chords

*This article delves into different geometric components intrinsic to circles: tangents, secants, and chords.*

## The main components of a circle

Whenever we want to distinguish a circle uniquely, the parameters conventionally used are:

- Diameter – The distance between two points on the circle passing through the center. This is also considered the longest straight line distance between two points on the boundary of a circle.
- Radius – The radius is the distance from the center of a circle to any point on the boundary.
- The circumference – The circumference is the length/perimeter of a circle’s boundary. The circumference is directly proportional to the diameter of a circle, which means the diameter and the corresponding circumference of a given circle always maintain a constant ratio. This ratio is what is famously known as “pi”.

All of the above parameters allow us to understand the size of a circle, and distinguish them from other non-congruent circles.

Various other geometrical elements originate through the existence of objects with curvature such as the circle. Let’s find out more about three such elements.

## The Tangent

A Tangent is a component that exists in objects or elements with a curvature. A tangent is a line drawn such that it intersects a single point on a curved shape or surface. The diagram below depicts a tangent to a circle intersecting at a single point marked P.

There are various characteristics associated with tangents:

- Uniqueness: For every point on the circle’s boundary, there is exactly one tangent.
- Orthogonality: At the point of contact, the tangent makes a right angle with the radius.
- External Tangents: There are exactly two tangents that can be drawn from a given external point, each touching the circle at one distinct point.

## The Secant

The Secant is an object similar to a tangent, in that it is a line that exists as a by-product of objects with curvature, like circles. A secant is a line that intersects a circle at exactly two distinct points as shown in the diagram below.

The word secant originates from the Latin word “secare” which means to cut. This is intuitive as a secant cuts a circle and splits it into two segments.

Some characteristics of the secant line are:

- There are infinitely many secant lines that can be drawn such that they all pass through a common point on the circumference.
- A secant line is identical to another secant line if and only if both secant lines share two points of intersection.
- A secant line that passes through the center of a circle splits the circle into two equal segments, i.e. semicircles.

Finally, a geometrical element that can be thought of as being derived from a secant is the chord.

## Chords

The only difference between a secant and a chord is that a chord does not extend beyond the boundaries of a circle, unlike a secant. In other words, a chord is a line segment with finite length, whereas a secant is a line that extends infinitely in both directions.

A chord can be described as a line segment that joins two distinct points in the circumference of a circle.

Some geometrical properties of a chord would be:

- Each chord divides the circle into two segments.
- The longest chord of a circle is obtained when the line segment passes through the center of the circle and has a length equivalent to the diameter of the circle.
- The perpendicular bisector of any chord always passes through the center of a circle. (Illustrated in the diagram above).

## Conclusion

Understanding tangents, secants, and chords not only enhances our grasp of circle geometry but also our ability to apply these concepts to various practical and theoretical contexts. This exploration reveals the extensive role that simple geometric forms can play in our understanding of the world around us.