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Tangent of a Circle – Meaning, Properties, Examples

tangent of a circle

This post is also available in: हिन्दी (Hindi)

The word ‘tangent’ is derived from the Latin word ‘tangere’ meaning ‘to touch’. The line that intersects the circle exactly at one point on its circumference and never enters the circle’s interior is a tangent. In other words, we can say that a line that just touches the circle is called the tangent of a circle.

Let’s understand what is a tangent of a circle,  and its properties with examples.

What is a Tangent of a Circle?

A tangent of a circle is the line that touches the circle at only one point. You can draw only one tangent at a point to a circle. The point at which the tangent touches the circle is called the point of tangency(or the point of contact). The tangent and the radius of a circle are perpendicular to each other at the point of tangency(or point of contact).

tangent of a circle

In the above figure, the line segment $\text{AB}$ is a tangent to the circle with centre $\text{O}$. The radius of the circle $\text{OP}$ is perpendicular to the tangent $\text{AB}$, i.e., $\angle \text{APO} = \angle \text{BPO} = 90^{\circ}$.

Let’s now prove that the tangent $\text{AB}$ is perpendicular to the radius $\text{OP}$.

tangent of a circle

From the above discussion, it can be concluded that:

Let’s now draw a few more line segments from the centre $\text{O}$ on the tangent, meeting at say $\text{B}$ and $\text{C}$.

Note that the points $\text{A}$, $\text{B}$, $\text{C}$ and $\text{D}$ all lie outside the circle (are exterior points), whereas the point $\text{P}$ lies on the circle.

Therefore, $\text{OA} \gt \text{OP}$, $\text{OB} \gt \text{OP}$, $\text{OC} \gt \text{OP}$, and $\text{OD} \gt \text{OP}$, as the distance of any external point is always greater than the distance of a point lying on a circle from the centre of the circle.

Since $\text{OP}$ is the shortest among all the line segments, therefore, $\text{OP} \perp \text{AB}$.

From the above discussion, it can be concluded that

  • The tangent touches the circle at only one point
  • We can call the line containing the radius through the point of contact as ‘normal’ to the circle at the point

Note: The tangent to a circle is a special case of the secant when the two endpoints of its corresponding chord coincide.

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Tangent Properties

The tangent has three important properties:

  • A tangent touches a circle at only one point.
  • A tangent is a line that never enters the circle’s interior.
  • The tangent touches the circle’s radius at the point of tangency at a right angle.

Apart from the above-listed properties, a tangent to the circle has mathematical theorems associated with it and those theorems are used while doing major calculations in geometry.

How Many Tangents Can Be Drawn From a Point to a Circle?

Let’s draw a circle and take a point P inside it. Can you draw a tangent to the circle through this point? You will find that all

the lines through this point intersect the circle in two points. So, it is not possible to draw any tangent to a circle through a point inside it.

tangent of a circle

Next, take a point $\text{P}$ on the circle and draw tangents through this point. You have already observed above that there is only one tangent to the circle at such a point.

tangent of a circle

Finally, take point $\text{P}$ outside the circle and try to draw tangents to the circle from this point. What do you observe? You will find that you can draw exactly two tangents to the circle through this point.

tangent of a circle

Therefore, the following three cases are observed:

  • There is no tangent to a circle passing through a point lying inside the circle.
  • There is one and only one tangent to a circle passing through a point lying on the circle.
  • There are exactly two tangents to a circle through a point lying outside the circle.

Two Tangents Theorem

Let’s consider two tangents drawn to a circle from an exterior point $\text{C}$. Let the points of contact be $\text{A}$ and $\text{B}$, as shown in the figure below.

tangent of a circle

The theorem states that The lengths of tangents drawn from an external point to a circle are equal.

Let’s now prove the above theorem. For this let’s join $\text{C}$ with $\text{P}$

In $\triangle \text{OAC}$ and $\triangle \text{OBC}$,

$\text{OA} = \text{OA}$ (Radii of a circle)

$\text{OC} = \text{OC}$ (Side common to both the triangles)

$\angle \text{OAC} = \angle \text{OBC} = 90^{\circ}$ (Angle between tangent and radius at the point of contact)

Therefore, $\triangle \text{OAC} \cong \triangle \text{OBC}$.

Thus $\text{AC} = \text{BC}$ (Corresponding Parts of Congruent Triangles).

Famous Math Competitions for Kids

Practice Problems

  1. What is a tangent of a circle?
  2. What is a secant of a circle?
  3. How many tangents can be drawn from a point to a circle when the point is lying
    • inside the circle
    • on the circle
    • outside the circle
  4. What is the measure of an angle between the radius and the tangent of a circle at the point of contact?

FAQs

What is a tangent of a circle?

A tangent is a line that touches the circle at only one point and never enters the circle’s interior. A tangent and the radius of a circle are perpendicular to each other at the point of contact.

What are the two major theorems of a tangent to circle?

The two major theorems of a tangent to circle theorems are 
a) The tangent at any point of a circle is perpendicular to the radius through the point of contact.
b) The lengths of the two tangents drawn from an external point to a circle are equal.

What are the properties of tangents to a circle?

The major properties of a tangent to a circle are
a) The tangent is a straight line that touches the circle at only one point.
b) It is perpendicular to the radius at the point of tangency.
c) It never enters the circle’s interior.
d) The lengths of two tangents to a circle from the same external point are equal.

Conclusion

A tangent of a circle is the line that touches the circle at only one point.  You can draw only one tangent at a point to a circle and this point is known as the point of tangency(or the point of contact). One cannot draw a tangent from a point lying inside the circle and two tangents of equal lengths can be drawn from a point lying outside the circle.

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