Number of Tangents from a Point on a Circle: The collection of all points in a plane at a constant distance from a fixed point is called a circle. This constant distance is the radius of a circle, and the fixed point is called the centre.

A secant is a line that intersects a circle in two distinct points. A tangent to the circle at a point is a line touching the circle only at one point.

In this article, we will study the number of tangents from a given point on the circle and solve some example problems on the same.

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Tangent

Observe the below figure where the line \(PQ\) is passing through a single point \(A\) on the circle.
The only point \(A\) is familiar to the line \(PQ\) and the circle. Here, the line \(PQ\) is called the tangent to the circle.

As a result, a tangent to a circle is a line that intersects the circle exactly once. This is known as the tangent’s point of contact, and it is at this point, the line is said to meet the circle.

Tangent comes from the Latin word tangere, which means “to touch.”

The only point on the tangent common to both the tangent and the circle is the point of contact; all other points on the tangent are outside the circle. As a result, among all the locations on a tangent to a circle, the point of contact is the one that is closest to the circle’s centre.

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

Know Tangent of a Circle

Properties of a Tangent to a Circle

i. The tangent makes \(90\) degrees to the radius through the point of contact at any point on a circle.
ii. The length of tangents taken from an exterior point to a circle is the same.
iii. When two tangents are drawn from an exterior point, the tangents at the centre subtend equal angles.
iv. In two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact.
v. The tangents drawn at the endpoints of a circle’s diameter are parallel.
vi. There is only one tangent at any point along the circumference of a circle.
vii. The sum of the angle between two tangents drawn from an exterior point to a circle and the angle subtended by the segments joining the contact points to the centre is \({180^ \circ }.\)
viii. The perpendicular at the contact point of the tangent to a circle passes through the centre.

When you look at a bicycle’s wheels, you’ll see that all of the spokes are aligned along the radii. A bicycle’s wheels move along lines tangents at the spots where they touch the ground when it runs. You’ll observe that the radii across the contact points with the ground appear to be right angles to the tangent in all orientations.

We will prove this property and some other properties of tangents to a circle as theorems.

Theorems on Properties of Tangent to a Circle

Theorem 1: A tangent to a circle is perpendicular to the radius through the point of contact.
Given: A circle \(C\left({O,r} \right)\) and a tangent \(AB\) at a point \(P.\)
To prove \(OP \bot AB.\)
Construction: Take any point \(Q,\) other than \(P,\) on the tangent \(AB.\) Join \(OQ.\) Suppose \(OQ\) meets the circle at \(R.\)
Proof: We know that among all line segments joining the point \(O\) to a point on \(AB,\) the shortest one is perpendicular to \(AB.\) So, to prove that \(OP \bot AB,\) it is sufficient to prove that \(OP\) is shorter than any other segment joining \(O\) to any point of \(AB.\)
\(OP = OR\) (Radii of the same circle)
Now, \(OQ = OR + RQ\)
\( \Rightarrow OQ > OR\)
\( \Rightarrow OQ > OP\) \(\left({OP = OR} \right)\)
\( \Rightarrow OP < OQ\)
Thus, \(OP\) is shorter than any other segment joining \(O\) to any point of \(AB.\)
Hence, \(OP \bot AB\)

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Theorem 2: A line drawn through the endpoint of a radius and perpendicular to it is a tangent to a circle.
Given: A circle \(C\left({O,r} \right)\) with radius \(OP\) and a line \(APB,\) perpendicular to \(OP.\)
To prove: \(AB\) is a tangent to the circle at the point \(P.\)
Proof: Take a point \(Q,\) different from \(P,\) on the line \(AB.\)
Now, \(OP \bot AB\)
\( \Rightarrow \) Among all line segments joining \(O\) to a point on \(AB,OP\) is the shortest.
\( \Rightarrow OP < OQ\)
\( \Rightarrow OQ > OP\)
\( \Rightarrow Q\) lies outside the circle.
Thus, every point on \(AB,\) other than \(P,\) lies outside the circle. This shows that \(AB\) meets the circle only at the point \(P.\)
Hence, \(AB\) is a tangent to the circle at \(P.\)
Let’s do the following activity to understand how many tangents there are from a point on a circle.

Activity to Find the Number of Tangents From a Point to a Circle

Draw a circle on the paper and take a point \(P\) inside it. Let us now try to draw a tangent to the circle through this point \(P.\) We observe that all the lines through point \(P\) intersect the circle in two distinct points. Therefore, none of them can be a tangent to the circle, as shown below.
Now, take a point \(P\) on the circle. We know that there exists one and only one tangent to a circle at a given point on it. So, there is only one tangent to the circle at point \(P\) as shown below.
Finally, let us take point \(P\) outside the circle and draw tangents to the circle from point \(P.\) We observe that we can draw precisely two tangents to the circle through point \(P\) as shown below.

Number of Tangents from a Point on a Circle

i. No tangents are passing via a point inside the circle.
ii. There is only one tangent that passes through a circle’s point.
iii. Only two tangents pass via a location outside of a circle. \(PA\) and \(PA’\) are two tangents from a point \(P\) outside the circle in the diagram below.
In the below figure, \(P{T_1}\) and \(P{T_2}\) are two tangents drawn from a point \(P\) lying outside the circle. These tangents touch the circle at \({T_1}\) and \({T_2},\) respectively. So, \({T_1}\) and \({T_2}\) are known as contact points of tangents \(P{T_1}\) and \(P{T_2},\) respectively.

Length of the Tangent

The length of the tangent segment between the point and the given point of contact with the circle is called the length of the tangent from the point to the circle.

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In the below figure, \(P{T_1}\) and \(P{T_2}\) are the lengths of tangents from point \(P\) to the circle. In the following theorem, we will prove that these two lengths are equal.
Theorem 3: The lengths of two tangents drawn from an external point to a circle are equal.
Given: \(AP\) and \(AQ\) are two tangents from a point \(A\) to a circle \(C\left({O,r} \right).\)
To prove: \(AP = AQ\)
Construction: Join \(OP,OQ\) and \(OA\)
Proof: To prove that \(AP = AQ,\) we shall first prove that \(\Delta OPA \cong \Delta OQA.\)
Since a tangent at any point of a circle is perpendicular to the radius through the point of contact.
Therefore, \(OP \bot AP\) and \(OQ \bot AQ\)
\( \Rightarrow \angle OPA = \angle OQA = {90^ \circ }…….\left({\text{i}} \right)\)
Now, in right-angled triangles \(OPA\) and \(OQA,\) we have
\(OP = OQ\) (Radii of a circle)
\(\angle OPA = \angle OQA\) (From \(\left({\text{i}} \right)\)) and,
\(OA = OA\) (Common)
So, by the \({\text{RHS}}\) criterion of congruence, we get
\(\Delta OPA \cong \Delta OQA\)
\( \Rightarrow AP = AQ\)
Theorem 4: If two tangents are drawn from an external point then,
1. The tangents subtend equal angles at the centre, and
2. The tangents are equally inclined to the line segment joining the centre to that point
Given: A circle \(C\left({O,r} \right)\) and a point \(A\) outside the circle such that \(AP\) and \(AQ\) are the tangents drawn to the circle from point \(A.\)
To prove: \(\angle AOP = \angle AOQ\) and \(\angle OAP = \angle OAQ\)
Proof: In right triangles \(OAP\) and \(OAQ,\) we have
\(AP = AQ\) (Tangents from external points are equal)
\(OP = OQ\) (Radii of the circle) and,
\(OA = OA\) (Common)
So, by the SSS criterion of congruence, we have
\(\Delta OAP \cong \Delta AQ\)
\( \Rightarrow \angle AOP = \angle AOQ\) and \(\angle OAP = OAQ\)

Solved Examples – Number of Tangents from a Point on a Circle

Q.1. A point P is \(13\,{\text{cm}}\) from the centre of the circle. The length of the tangent drawn from point P to the circle is \(12\,{\text{cm}}.\) Find the radius of the circle.

Ans: Since the tangent to a circle is perpendicular to the radius through the point of contact.
Therefore, \(\angle OPT = {90^ \circ }\)
In right-angled triangle \(OTP,\) we have
\(O{P^2} = O{T^2} + P{T^2}\)
\( \Rightarrow {13^2} = O{T^2} + {12^2}\)
\( \Rightarrow O{T^2} = {13^2} – {12^2} = \left({13 – 12} \right)\left({13 + 12} \right) = 25\)
\( \Rightarrow OT = 5\)
Hence, the radius of the circle is \(5\,{\text{cm}}.\)

Q.2. Two tangents TP and TQ are drawn to a circle with a centre O from an external point T. Prove that \(\angle PTQ = 2\angle OPQ\)

Ans: We are given a circle with centre \(O,\) an external point \(T,\) and two tangents \(TP\) and \(TQ\) to the circle, where \(P,Q\) are the contact points. We need to prove that \(\angle PTQ = 2\angle OPQ.\)
Let \(\angle PTQ = \theta \)
We know that \(TP = TQ.\) So, \(TPQ\) is an isosceles triangle.
Therefore, \(\angle TPQ = TQP = \frac{1}{2}\left({{{180}^ \circ } – \theta } \right) = {90^ \circ } – \frac{\theta }{2}\)
Also, \(\angle OPT = {90^ \circ }\)
So, \(\angle OPQ = \angle OPT – \angle TPQ = {90^ \circ } – \left({{{90}^ \circ } – \frac{\theta }{2}} \right)\)
\( \Rightarrow \angle OPQ = \frac{\theta }{2}\)
\( \Rightarrow \angle PTQ = 2\angle OPQ\)

Q.3. In the below figure XP and XQ are tangents from X to the circle with a centre O. R is a point on the circle. Prove that \(XA + AR = XB + BR\)

Ans: Since lengths of tangents from an exterior point to a circle are equal
Therefore, \(XP = XQ……..\left({\text{i}} \right)\)
\(AP = AR \ldots \ldots \left({{\text{ii}}} \right)\)
\(BQ = BR \ldots \ldots \left({{\text{iii}}} \right)\)
Now, \(XP = XQ\)
\( \Rightarrow XA + AP = XB + BQ\)
\( \Rightarrow XA + AR = XB + BR\) (Using \(\left({\text{i}} \right)\) and \(\left({\text{ii}} \right)\))

Q.4. Find the tangent length drawn from a point whose distance from the centre of a circle is \(25\,{\text{cm}},\) given that the radius of the circle is \(7\,{\text{cm}}.\)
Ans: Let \(P\) be the given point, \(O\) be the centre of the circle, and \(PT\) be the length of the tangent from \(P.\) Then, \(OP = 25\,{\text{cm}}\) and \(OP = 7\,{\text{cm}}.\)

Since tangent to a circle is always perpendicular to the radius through the point of contact.
Therefore, \(\angle OTP = {90^ \circ }\)
In a right-angled triangle \(OTP,\) we have
\(O{P^2} = O{T^2} + P{T^2}\)
\( \Rightarrow {25^2} = {7^2} + P{T^2}\)
\( \Rightarrow P{T^2} = {25^2} – {7^2}\)
\( \Rightarrow PT = 24~{\text{cm}}\)
Hence, the length of the tangent from \(P = 24~{\text{cm}}\)

Q.5. In the below figure, if \(AB = AC,\) prove that \(BE = EC\)

Ans: Since tangents from an exterior point to a circle are equal in length.
Therefore, \(AD = AF……..\left({\text{i}} \right)\)
\(BD = BE……..\left({{\text{ii}}} \right)\)
\(CE = CF……..\left({{\text{iii}}} \right)\)
Now, \(AB = AC\)
\( \Rightarrow AB – AD = AC – AD\) (Subtracting \(AD\) from both sides)
\( \Rightarrow AB – AD = AC – AF\) (Using \(\left({\text{i}}\right)\))
\( \Rightarrow BD = CF\)
\( \Rightarrow BE = CF\) (Using \(\left({\text{ii}}\right)\))
\( \Rightarrow BE = CE\) (Using \(\left({\text{iii}}\right)\))

Summary

In this article, we have studied the meaning of tangent, properties of a tangent, theorems on properties of a tangent, number of tangents from a point to a circle, activity to find the number of tangents from a point to a circle and length of the tangents. Also, we have solved some example problems based on the number of tangents from a point to a circle.

Frequently Asked Questions (FAQ) – Number of Tangents from a Point on a Circle

Q.1. How many tangents can be drawn to a circle from a point on a circle?
Ans: No tangent line can be traced through a point within a circle. However, two tangent lines to a circle can be traced from a point outside the circle. There is only one tangent that passes through a circle’s point.

Q.2. How do you find the number of tangents to a circle?
Ans: Tangent is a line that only intersects a circle at one point. As a result, a circle can have an endless number of tangents. Also, we can draw two tangents from a point outside the circle.

Q.3. How many circles can be drawn passing through a point?
Ans: Passing through a given point, we can draw an infinite number of circles.

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

Q.5. What is the length of the tangent?
Ans: The length of the tangent is the segment between the external point, and the point of contact with the circle is called the length of the tangent.

We hope you find this article on ‘Number of Tangents from a Point on a Circle‘ helpful. In case of any queries, you can reach back to us in the comments section, and we will try to solve them.