Tangent of a Circle: Definition, Properties and Examples

We notice numerous circular objects in our environment, such as a circular clock, coins, frisbees, train wheels on the track, and so on. A Tangent of a Circle is a line that touches the circle’s boundary at exactly one point. The tangential point is the place where the line and the circle meet.

The word “tangent” is derived from the Latin word “tangere” (which means “to touch”), which was coined by a Danish mathematician named ‘Thomas Fineko’ in the early 1800s (1583). Using examples, this article will describe a tangent to a circle and illustrate its qualities.

In this article, we will dive deep into the concept of Tangent of a Circle, Definition, Properties, Examples, etc. Continue reading to know more.

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Tangent of a Circle Definition

In our daily life, we observe that two lines either intersect in a plane or not intersect at all. For example, two parallel lines in a railroad track never intersect each other, whereas, in windows grill design, the grills intersect each other. Do you know what happens if a curve and a line are given in a single plane? The curve can be a parabola, a circle, or any general curve. 

Similarly, what will happen if we consider the intersection of a line and a circle? We may get three situations as given in the following diagrams.

Definition: A tangent to a circle is a line that touches the circle at only one point. And the point of contact is known as the point of tangency.

Here, \(AB\) is the centre of the circle, and \(P\) is the point of tangency.

Real-life Examples of Tangent of a Circle

(i) When a cycle moves along a road, then the road becomes the tangent at each point when the wheels roll on it. 

(ii) When a stone is tied at one end of a string and rotated from the other end, the stone will follow a circular path. If we suddenly stop the motion, the stone will go in a direction tangential to the circular motion.

(iii) Abstract composition of yellow ball and curb, placed tangentially, with the projection of spherical shadow, on a stone background is also an example of a tangent of a circle.

Uses of Tangent of a Circle

Circles and tangent lines can be helpful in many real-world applications and fields of study, such as construction, landscaping, and engineering.

Number of Tangents From a Point on a Circle

Draw a circle on paper. Take a point \(P\) inside the circle. Can a tangent be drawn to the circle through this point \(P?\) 

We see that all the lines through this point \(P\) intersect the circle at two distinct points. So, it is impossible to draw any tangent to a circle from a point inside the circle.

Now, take a point \(P\) on the circle and draw tangents through this point. We observe that only one tangent can be drawn at any point on the circle. 

Finally, take a point \(P\) outside the circle and try to draw tangents to the circle from this point. What do we observe? We find that two tangents can be drawn to the circle through this point \(P.\)

The length of the tangent segment from the external point \(P\) and the point of contact with the circle is called the length of the tangent from the point \(P\) to the circle.

\(PA\) and \(PB\) are the lengths of the tangents from \(P\) to the circle.

Tangent of a Circle Method

Case 1: To draw only one tangent line 

Consider a circle with a centre \(O\) and draw a line perpendicular to the circle’s radius from a point on the circle. That perpendicular line is called the tangent to the circle.

Case 2: To draw two tangent lines 

Consider a circle with a centre \(O\) and draw two lines perpendicular to the circle’s radius from two distinct points on the circle. That perpendicular lines are called the tangent to the circle.

Properties of Tangent of a Circle

Theorem 1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Given: A circle \(C\left( {0,\,r} \right)\) and a tangent \(I\) at point \(A\).

Let us try to prove \(OA \bot I\)(\(OA\) is perpendicular to \(I\))

Construction: Take a point \(B,\) other than \(A,\) on the tangent \(I.\) Join \(OB.\) Suppose \(OB\) meets the circle at point \(C\).
\(OA=OC\)  (Radius of the same circle)
Now, \(OB = OC + BC.\)
\(∴ OB > OC\)
\(OB > OA\)
\(OA < OB\)
B is an arbitrary point on the tangent \(I\). Thus, \(OA\) is shorter than any other line segment joining \(O\) to any point on \(I\).

Here, \(OA \bot I\), the converse of the tangent theorem.

Theorem 2: A line perpendicular to the radius at its point on the circle is a tangent to the circle.

Given:  \(M\) is the centre of a circle, and \(MN\) is the radius. 
Line \(l \bot\) seg \(MN\) at \(N\)

To Prove: Line \(l\) is a tangent to the circle. 

Proof: Take any point \(P\), other than \(N\), on the line \(l\). Draw seg \(MP\).
Now in \(\Delta MNP,\angle N\) is a right angle.
Therefore, seg \(MP\) is the hypotenuse.
Therefore, \(segMP > {\mathop{\rm seg}\nolimits} MN\)
As seg \(MN\) is the radius, point P cannot be on the circle.
So, no other point except point \(N\), of line \(l\) is on the circle. 
Line \(l\) intersects the circle in only one point \(N\).

Therefore, line \(l\) is tangent to the circle.

Theorem 3: The lengths of tangents drawn from an external point to a circle are equal.

Given: a circle with centre \(O\), a point \(P\) lying outside the circle and two tangents \(PA, PB\) on the circle from \(P\).

To Prove: \(PA=PB\)

Construction: Join \(OA, OB\) and \(OP\). 

Proof: \(\angle OAP = \angle OBP = {90^ \circ }\) (Because a tangent at any point of a circle is perpendicular to the radius through the point of contact)
Now in right \(\Delta OAP\) and \(\Delta OBP\)
\(OA=OB\) (Radii of the same circle) 
\(OP=OP\) (Common) 
Therefore, \(\Delta OAP \cong \Delta OBP\) (By RHS congruency criteria) 

Hence, \(PA=PB\) (By CPCT)

Solved Examples – Tangent of a Circle

Q.1. In the given figure, a circle with centre \(D\) touches the sides of \(\angle ACB\) at \(A\) and \(B\). If \(\angle ACB = {52^ \circ }\), find the measure of \(\angle ADB\).
Ans: We know that the sum of all angles of a quadrilateral is \({360^ \circ }\)

\(\angle ACB + \angle CAD + \angle CBD + \angle ADB = {360^ \circ }\)
\(\Rightarrow {52^ \circ } + {90^ \circ } + {90^ \circ } + \angle ADB = {360^ \circ }\) (By using the tangent theorem)
\( \Rightarrow \angle ADB + {232^ \circ } = {360^ \circ }\)
\(\therefore \angle ADB = {360^ \circ } – {232^ \circ } = {128^ \circ }\)

Q.2. Find the length of the tangent drawn from a point whose distance from the centre of a circle is \({\rm{5}}\,{\rm{cm}}\) and radius of the circle is \({\rm{3}}\,{\rm{cm}}{\rm{.}}\)
Ans:

Given \(OP = 5\;{\rm{cm}}\),radius \(r = 3\;{\rm{cm}}\)
In the right-angled \(\Delta OTP\)
\(O{P^2} = O{T^2} + P{T^2}\) (by Pythagoras theorem) 
\({(5)^2} = {(3)^2} + P{T^2}\)
\( \Rightarrow P{T^2} = {(5)^2} – {(3)^2} = 25 – 9 = 16\)
\(\Rightarrow PT = 4\;{\rm{cm}}\)
Hence, the length of the tangent \(PT = 4\;{\rm{cm}}.\)

Q.3. If radii of two concentric circles are \(4\,{\rm{cm}}\) and \(5\,{\rm{cm}}\) respectively, then find the length of the chord of the bigger circle, which is a tangent to the smaller circle. 
Ans:

Given, \(OA = 4\;{\rm{cm}},OB = 5\;{\rm{cm}}\)
also, \(OA \bot BC\)
\(O{B^2} = O{A^2} + A{B^2}\)
\({(5)^2} = {(4)^2} + A{B^2}\)
\( \Rightarrow A{B^2} = {(5)^2} – {(4)^2} = 25 – 16 = 9\)
\( \Rightarrow AB = 3\;{\rm{cm}}\)
Therefore, \(BC = 2AB = 2 \times 3 = 6\;{\rm{cm}}\)

Q.4. \(\Delta ABC\) is circumscribing a circle in the given figure. Find the length of the side \(BC\). 

\(AN = AM = 3\;{\rm{cm}}\) (Tangents drawn from same external point are equal)
\(BN = BL = 4\;{\rm{cm}}\)
\(CL = CM = AC – AM = 9 – 3 = 6\;{\rm{cm}}\)
\( \Rightarrow BC = BL + CL = 4 + 6 = 10\;{\rm{cm}}\)

Q.5. In the given figure, two circles touch each other at point \(C\). Prove that the common tangent to the circle at \(C\), bisects the common tangent at \(P\) and \(Q\).

Solution: We know that the tangents drawn from an external point to a circle are equal.
Therefore, \(RP=RC\) and \(RC=RQ\)
\(⇒RP=RQ\)
\(⇒R\) is the midpoint of \(PQ\).

Summary

In this article, we discussed that the tangent to a circle is a line that touches the circle at exactly one point. With the help of real-life applications, one can easily relate to the concept. We also discussed properties related to the tangent to a circle, the number of tangents drawn from a point lying inside the circle, on the circle and outside the circle. 

With the help of this, one can easily understand the difference between a chord and a tangent line.

Frequently Asked Questions

We have provided some frequently asked questions about tangent of a circle here:

Q.1. What is the point of tangency of a circle?
Ans: The point of contact of the circle and the tangent line is called the point of tangency of a circle.

Q.2. What is/are the number of tangents drawn from a point outside the circle?
Ans: When a point lies outside the circle, two tangents can be drawn from that point to the circle.

Q.3. What is/are the number of tangents drawn from a point inside the circle?
Ans: When a point lies inside the circle, no tangent can be drawn. 

Q.4. What is a tangent of a circle?
Ans: A tangent to a circle is a line that intersects the circle at only one point.

Q.5. Does radius bisect a tangent?
Ans: No, it does not bisect the tangent, but it is perpendicular to the tangent through a point of contact.

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