Circles Class 10 Maths Chapter 10 CBSE Notes - 2025-26

1. Tangent to a Circle: A tangent to a circle is a straight line that only touches the circle once. The point of tangency is the name given to this location. At the point of tangency, the tangent to a circle is perpendicular to the radius.

2. Non-intersecting lines are made up of two or more lines that do not intersect. In fig (i), the circle and the line AB have no common point. It's worth noting that: 

• Lines that do not intersect can never meet.

• The parallel lines are another name for them.

• They stay at the same distance from one another at all times. 

3. A secant is a line that crosses a curve at two or more separate locations. A secant intersects a circle at exactly two locations in the case of a circle. In fig (ii), the line AB intersects the circle at two points A and B. AB is the secant of the circle. 

5. Figure (iii): The line AB only touches the circle at one place. P denotes a point on a line and a point on a circle. The point of contact is denoted by the letter P. The tangent to the circle at P is AB.

5. Number of Tangents from a Point on a Circle

There are no tangents to the circle that can be made from a point inside the circle.

Only one tangent to a circle can be traced from a point on the circle.

P is a point on the circle in this illustration. At P, there is just one tangent. The point of contact is denoted by the letter P.

Two tangents to a circle can be made from a point outside the circle. P is the exterior point in this diagram. The tangents to the circle at points Q and R are PQ and PR, respectively. The length of a tangent is the distance between the exterior point and the point of contact of the tangent's segment. PQ and PR are the lengths of the two tangents in this diagram.

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

Given: 

A tangent to the circle with centre O is AB. The point of contact is denoted by the letter P. The radius of the circle is denoted by OP.

To prove: \[\text{OP}\bot \text{AB}\]

Proof:

Let Q be any point on the tangent AB other than P, outside the circle.

For any tangent point Q that is not P.

The shortest distance between point O and line AB is OP.

The theorem is therefore proved by \[\text{OP}\bot \text{AB}\] (The shortest line segment drawn from a point to a given line is perpendicular to the line).

As a Result of the Preceding Theorem,

• The point of contact is crossed by the perpendicular drawn from the centre to the tangent of a circle.
  OP is the radius of the circle with centre O. The tangent to the circle at P is the perpendicular OP which is drawn at P.

• Theorem 2: The lengths of tangents drawn from an external point to a circle are equal.

Given: P is the outermost point of a circle with the centre O. The tangents from P to the circle are PA and PB. The points of contact are A and B.

To prove:

\[\text{PA=PB}\]

Construction:

Join OA, OB, OP.

Proof:

In \[\text{ }\!\!\Delta\!\!\text{ APO and  }\!\!\Delta\!\!\text{ BPO}\],

$\text{OA=OB}$, radius of the same circle.

$\text{OP=OP}$, common side

\[\text{PA=PB}\], by CPCT theorem, third side of the triangles

According to the following theorem, 

1. (CPCT) This indicates that near the circle's centre, the two tangents subtend equal angles. 

2. (CPCT) The tangents to the line connecting the point and the circle's centre are both equally inclined.

Alternatively, the circle's centre can be found on the angle bisector of $\Delta \text{APB}$, hence, \[\text{PA=PB}\].

Class 10 Maths Chapter 10 Circles Revision Notes

Download Class 10 Maths Chapter 10 Circles Revision Notes  Free PDF

As the saying goes - practice makes a man perfect, we have designed these Circles Class 10 Notes in pdf format  to help you sort out your learning preferences. The pdf can be downloaded free of cost through the  direct link provided on this page.

Here at Vedantu, we provide these Circles Class 10 notes PDF for you to practice and score well in your examinations. To help most students prepare for this topic and score good marks in their Maths examination, Circles Notes can be downloaded simply by clicking once on the pdf link given below. Download the PDF to get complete information about circles.

Topics Covered in Class 10 Maths Chapter 10 Circles

Tangent to a Circle

A tangent is a line touching a circle at one point. 

1. Non-intersecting line - fig (i): The circle and the line AB have no common point.

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2. Secant - fig (ii): The line AB intersects the circle at two points A and B. AB is the secant of the circle.

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3. Tangent - fig (iii): The line AB touches the circle at only one point. P is the point on the line and on the circle. P is called the point of contact. AB is the tangent to the circle at P.

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Number of Tangents from a Point on a Circle

1. From a point inside a circle, no tangents can be drawn to the circle.

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2. From a point on a circle, only 1 tangent can be drawn to the circle. In this figure, P is a point on the circle. There is only 1 tangent at P. P is called the point of contact.

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3. From a point outside a circle, exactly 2 tangents can be drawn to the circle. In this figure, P is the external point. PQ and PR are the tangents to the circle at points Q and R respectively. The length of a tangent is the length of the segment of the tangent from the external point to the point of contact. In this figure, PQ and PR are the lengths of the 2 tangents.
   
   

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Circle Chapter Class 10 Notes: An Overview

Here is a quick overview of all the basic definitions covered in the chapter:

Circle

It is defined as collecting all the points in a defined plane that are placed at a constant distance to a fixed point. 

Centre

The fixed point from which all other points are at the same distance is called the centre. 

Radius

The fixed distance from the centre from which all other points are at an equal distance is called the radius. 

Chord

It is defined as the line segment which joins two points on a circle.

Diameter

It is the longest chord of the circle which passes through its centre. 

Tangent

The line which meets the circle at one point or two coincidences is defined as the tangent. The tangent on a circle is always perpendicular to the radius at the point of its contact. 

The length of the tangents drawn from an external point to a circle is equal. 

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In the above-given diagram, PA and PB are the tangents of the circle. Here, according to the above-mentioned property, PA=PB. 

Properties of Tangents drawn to a Circle

1. At one point of contact in a circle, there can be only a tangent present. 

2. It isn’t possible to draw tangent from any point outside the circle. 

3. From any point outside the circle, there are only two tangents that are present. 

Theorems proving the properties of the tangent to a circle

The Class 10 Maths Circles notes comprise of various theorems which include:

Theorem 1:

It states that the tangent passing through any point of the circle lies perpendicular to the radius through the point of contact. 

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Given:

In the above figure, XY is a tangent passing through the point P of the circle having centre O. 

To Prove:

OP⟂XY

Construction:

Construct a point Q on the tangent XY and join it with the centre O making OQ. 

Proof:

If the point Q lying on XY is inside the circle, then XY will form a secant and not a tangent. Therefore, OQ>OP. The same will be the case of all the points lying on the tangent XY. Therefore, OP will always be the shortest distance from O in all cases. 

So, OP⟂XY, making it the shortest side of the perpendicular as well.

Theorem 2:

If drawn through the endpoint of the radius, a line perpendicular to it will be a tangent to the given circle. 

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Given:

A circle with centre O and radius and the line APB lies perpendicularly to OP. Here OP is the radius of the circle. 

To Prove:

AB is the tangent of the circle. 

Construction:

Take a point Q lying on the line AB. Join it with the centre, forming OQ. This point should be different from Q. 

Proof:

Here, OP<OQ. 

This implies that the point Q lies outside the circle. 

Also, all the other points lying on the line AB will lie outside the circle. 

This implies that AB will meet the circle at P. 

Hence, this proves that AB is a tangent to the circle. 

Theorem 3:

It states that the tangents drawn from an external point to a circle are equal in length. 

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Given:

Here, PT and PS are the tangents drawn to the circle having centre O from an external point P.

To Prove:

PT=PS

Construction:

Join the points T and S to O, and the external point P to O. 

Proof:

Here is given triangles OTP and OSP, 

OT=OS (they both are radii of the given circle)

OP=OP (common sides of the triangle)

∠OTP = ∠OSP (Both are tangents to the circle, which are perpendicular, according to the theorem)

Therefore, Triangle OTP = OSP (By R.H.S. Property)

Therefore, PT= PS (By CPCT property)

Hence, Proved. 

In the notes of Circle Class 10, all the theorems related to tangents. These are followed by solutions to exercises. The chapter consisted of two exercises, covering all the questions related to circles and their concepts. These questions are mainly related to the properties and theorems of circles. The exercises also include miscellaneous questions which require special attention to be paid by the students. 

Solved Questions

Question 1.

In the figure given below, if AB and AC are both tangents to the circle with centre O such that ∠BAC = 40°, Then find ∠BOC. Read this article on revision notes for Class 10 Maths Chapter 10 on Circles to get your last-minute exam revision right. We have made things easy for you!

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Solution:

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AB and AC are tangents

Therefore, ∠ABO = ∠ACO = 90°

In ABOC,

∠ABO + ∠ACO + ∠BAC + ∠BOC = 360°

90° + 90° + 40° + ∠BOC = 360°

∠BOC = 360 – 220° = 140°

Question 2.

In the figure given below, the circle touches the side DF of AEDF at H and touches ED and EF produced at K and M, respectively. If EK = 9 cm, then determine the perimeter of AEDF (in cm). 

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Solution:

Perimeter of ∆EDF = 2(EK) = 2(9) = 18 cm.

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Conclusion:

The Class 10 Maths Chapter 10 Notes covers all the essential topics related to circles. Our experts have curated these notes to help the students practice well and score better in their examinations. The students can make good use of the PDF available and study whenever they wish. Furthermore, all these topics are covered with suitable examples.

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