Tangent of a Circle – Meaning, Properties, Examples

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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).

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 $\text{OP}$.

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.

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.

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.

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.

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.

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).

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

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.

Recommended Reading

• How to Construct a Perpendicular Line (With Steps & Examples)
• How to Construct Parallel Lines(With Steps & Examples)
• How To Construct a Line Segment(With Steps & Examples)
• What are Collinear Points in Geometry – Definition, Properties & Examples
• What is a Transversal Line in Geometry – Definition, Properties & Examples
• What are Parallel Lines in Geometry – Definition, Properties & Examples
• What is Concurrent lines in Geometry – Definition, Conditions & Examples
• What is Half Line in Geometry – Definition, Properties & Examples
• What is a Perpendicular Line in Geometry – Definition, Properties & Examples
• Difference Between Axiom, Postulate and Theorem
• Lines in Geometry(Definition, Types & Examples)
• What Are 2D Shapes – Names, Definitions & Properties
• 3D Shapes – Definition, Properties & Types