Tangent of a Circle

Let’s begin with refreshing our understanding of a circle. A circle is defined as a round plane figure with a circumference or boundary with points that are equidistant from the center. A key concept or feature of a circle is its radius. The radius of a circle is defined as the distance between any point of the circle’s circumference and center point.  The next key feature is the tangent of a circle. A tangent of a circle is defined as a line that intersects the circle’s circumference at only one point. This point where the line touches the circle is called the point of tangency.  

Understanding What Is Tangent of Circle

A tangent of a circle does not cross through the circle or runs parallel to the circle. It has to meet one point at the circumference in order to meet the criteria of a tangent. 

Let’s consider the three possibilities of any line and a circle:

1. In the diagram below, a line runs across two points on the circumference of the circle, point A and point B. Such a line is called the secant of the circle and the part of the circle under that line is called the chord of the circle. 

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2. Another scenario that can happen with a circle and a line is when the line runs parallel to the circle or does not have a touchpoint or any intersecting point. The diagram below depicts such a scenario.

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3. This diagram below depicts exactly what a tangent to a circle is. You can see from this diagram that a line touches a single point on the circumference of a circle. The point where the line touches the circle is called the point of tangency.

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Properties of Tangent of a Circle

By developing an understanding of tangent through the knowledge of its properties, one can solve any problem related to the tangent of a circle or other geometry related questions.

There are five major properties of the tangent of a circle which shall be discussed below.

1. The first property specifies that a line that is tangent to a circle is always perpendicular to the radius. Remember this property is corresponding to the point of tangency.

2. The second important property says that when two or more tangent lines a circle extends from an external point (P) then these tangent lines are also perpendicular to the radii corresponding to the point of tangency.

3. The third property mentions that when there are two intersecting lines to circle tangent then the center is always on the bisector of the angle formed by the tangent lines. 

4. Proceeding to the fourth property, the radii of any circle is always a perpendicular bisector of the segment defined by any two given points of that circle.

5. Lastly, the fifth property that shall be discussed here specifies that the centers of two tangent circles are aligned with the point of tangency.

It is important to be mindful of these properties when learning about the tangent of a circle. These properties definitely help in solving questions related to the tangent of a circle and at the same time can help overcome any confusion related to any question.

The Formula of Tangent of a Circle

Let’s consider there is a point A that lies outside a circle. From this point, A (point of tangency), draw two tangent lines touching two points P and Q respectively at the curve of the circle. Take two other points, X and Y, from which a secant is drawn inside the circle. The next step involves drawing a line that connects the secant to the tangent or point of tangency, and this line can be referred to as AZ. 

Therefore the relationship between this line connecting secant and point of tangency, secant, and tangents can be developed which is mentioned below:

\[\frac{AP}{AZ} = \frac{AZ}{AQ}\]

AZ2 = AP.AQ

This relationship above is generally used when solving for the tangent of a circle. If the length of the secant is provided, then a direct application of this formula can be put to use. Furthermore, this tangent of a circle formula can solve for a finding of the secant and the coordinate points of the point of tangency. 

This article summarises the definition of the tangent of a circle, the special properties, and the formula. Following the above-mentioned information, it can be helpful in developing a basic understanding of the circle, tangent of a circle, and other geometrical properties. Moreover, this concept is further elaborated when learning the coordinates of geometry or even derivation.