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Circles Class 10 Notes CBSE Maths Chapter 10 (Free PDF Download)

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Class 10 Maths Revision Notes for Circles of Chapter 10 - Free PDF Download

Providing the best help for our students has always been our priority. To help you understand the topics Circles better, our experts at Vedantu have come with an excellent Class 10 Maths Chapter 10 Circles Revision Notes with a view to make you understand the concepts of the topic clearly. 

The Circles Class 10 Notes have been prepared by expert teachers at Vedantu and have been designed to bring transparency to all the important topics given in this chapter. Students can further use these notes to practice questions  as well as to have a thorough revision of the chapter quickly before the exam without missing out on any important topic.

Vedantu is a platform that provides free CBSE Solutions (NCERT) and other study materials for students. You can download Class 10 Maths NCERT Solutions to help you to revise complete syllabus and score more marks in your examinations.

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Circles Class 10 Notes CBSE Maths Chapter 10 (Free PDF Download)
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Access Class 10 Maths Chapter 10 - Circles

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: 


Non-intersecting lines


  • 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. 


Secant


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.


Tangent

 

5. Number of Tangents from a Point on a Circle


Number of tangent lines from a point inside the circle

 

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

 

Tangents outside 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 of a circle
 

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.

 

Tangent Perpendicular to radius
 

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.

 

Tangents from external point to circle

 

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.


(image will be uploaded  soon)


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


(image will be uploaded  soon)

 

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.

 

(image will be uploaded  soon)

Number of Tangents from a Point on a Circle

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

 

(image will be uploaded  soon)

  1. 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.

 

(image will be uploaded  soon)

  1. 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.

(image will be uploaded  soon)

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. 

 

(image will be uploaded  soon)

 

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. 


(Image will be uploaded  soon)

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. 


(image will be uploaded  soon)


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. 

 

(Image will be uploaded  soon)

 

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!

 

(Image Will Be Updated Soon)

Solution:

(Image Will Be Updated Soon)

 

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). 

(Image Will Be Updated Soon)

 

Solution:

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

 

What are the Benefits of Referring to Vedantu’s Revision Notes for Class 10 Maths Chapter 10 - Circles

  • Provides quick, clear summaries of key concepts.

  • Simplifies complex topics for better understanding.

  • Efficient tool for last-minute exam prep.

  • Enhances retention of crucial information.

  • Supports effective exam preparation with key points and tips.

  • Saves time by consolidating information.

  • Prioritizes important topics and questions.

  • Offers practical examples for real-world connections.

  • Boosts student confidence for exams.


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.


Related Study Materials for Class 10 Maths Chapter 10 Circles


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FAQs on Circles Class 10 Notes CBSE Maths Chapter 10 (Free PDF Download)

1. What is the definition of a circle according to the Class 10 Maths syllabus?

A circle is defined as the collection of all points in a plane that are at a constant distance from a fixed point, called the centre. The constant distance is known as the radius as per the CBSE Class 10 Circles Revision Notes.

2. What are the key properties of tangents to a circle in Class 10 Circles Revision Notes?

The important properties of tangents in circles covered in Class 10 revision notes include:

  • A tangent touches the circle at exactly one point.
  • The tangent is always perpendicular to the radius at the point of contact.
  • From a point outside the circle, exactly two tangents can be drawn, and their lengths are equal.

3. How does the number of tangents vary with the position of a point relative to the circle?

  • No tangents can be drawn from a point inside the circle.
  • Exactly one tangent can be drawn from a point on the circle.
  • Exactly two tangents of equal length can be drawn from a point outside the circle.
These facts are repeatedly tested in CBSE Class 10 Maths board exams.

4. What is the main theorem on tangents stated in Class 10 Circles Notes?

The key theorem from Class 10 Circles Revision Notes is: The tangent to a circle at any point is perpendicular to the radius through the point of contact. This is fundamental for solving exam questions on circle tangents (CBSE 2025–26).

5. Why are the two tangents drawn from an external point to a circle always equal in length?

The lengths of two tangents drawn from an external point to a circle are always equal because the triangles formed (by joining the points of contact and the centre with the external point) are congruent by the RHS criterion. This property is proved using congruence in the chapter’s theorems and is key for reasoning questions in board exams.

6. Can a chord of a circle be a tangent? Why or why not?

No, a chord of a circle cannot be a tangent. A chord connects two points on the circle and passes through the interior, while a tangent touches the circle at only one point without entering the circle's interior. This conceptual distinction is important for MCQ and short-answer formats.

7. What are common misconceptions students have about tangents in Class 10 Circles?

Frequent misconceptions include:

  • Assuming that the lengths of tangents from different external points to the circle are equal (they are only equal when from the same external point)
  • Believing that tangents can be drawn from every point in the plane (not possible from interior points)
  • Confusing chords and tangents due to their line-like appearance
Recognizing these errors helps in avoiding common exam mistakes.

8. Give the differences between a secant and a tangent as per Class 10 Circles Notes.

  • Secant: A line that intersects a circle at exactly two distinct points.
  • Tangent: A line that touches a circle at exactly one point.
Understanding this difference is essential in both concept-based and application-based questions.

9. How do you prove that the radius at the point of contact is perpendicular to the tangent?

This is shown by demonstrating that the shortest distance from the centre of a circle to the tangent line is along the radius, and this shortest distance is perpendicular. Any other line from the centre to the tangent is longer than the radius, confirming that the radius and the tangent form a right angle at the point of contact.

10. Why is it important to learn all theorems related to tangents in Class 10 Circles Revision Notes?

These theorems form the backbone of both direct and application-based questions in CBSE board exams. A clear understanding allows students to solve unseen problems, prove properties, and tackle HOTS (Higher Order Thinking Skills) questions efficiently, as per CBSE 2025–26 requirements.

11. What key terms should be memorized from Class 10 Circles for efficient revision?

The most important terms are: circle, centre, radius, diameter, chord, secant, tangent, point of contact, perpendicular. Knowing these helps students write precise answers during exams.

12. How are revision notes useful for quick exam preparation of Class 10 Chapter 10 Circles?

Revision notes for Class 10 Circles offer:

  • Short summaries of concepts
  • Key theorems with proofs
  • Quick reference of formulas and properties
  • Definition highlights for one-mark questions
They consolidate information, making last-minute revision highly effective.

13. What is the relation between diameter and chord in the context of Class 10 Circles?

The diameter is the longest chord of a circle, passing through the centre. Every diameter is a chord, but not every chord is a diameter. Understanding this is frequently tested in objective-type questions.