CBSE Notes Class 10 Maths Chapter 10: Circles

1.0Introduction to circles 

A circle is a closed curve formed by all the points within a plane, which are all equidistant from any fixed central point. Therefore, some of the very important elements of a circle include the centre, radius, diameter, chord, and tangents.

This chapter centres on the properties and applications of tangents to a circle. These concepts are crucial to be studied not only for the purpose of board exams but also for higher studies in geometry. In order to help students grasp the material, these notes provide a summary of the chapter, including key ideas, definitions, formulae, and tips and tricks, in addition to solved examples. 

2.0CBSE Class 10 Maths Chapter 10 Circles - Revision Notes

Key Concepts Related to Circle 

Key concepts	Figure
1. Tangent to a circle:  A line intersecting the circle at only one point is known as a tangent to a circle. That is, there is only one tangent at a given point in a circle.
2. Secant of a circle  A secant is that line that cuts the circle through two points. Unlike the tangent, where it touches the circle at one point, a secant passes through the circle with two points.
3. Chord  A chord of a circle is any line segment whose endpoints are on the circle. Connecting two points on a circle, its longest chord is actually its diameter, which goes through the centre of the circle.
4. Point of contact  The point of contact is that point where a tangent touches the circle. At that point, the tangent is perpendicular to the radius drawn from the centre of the circle to the point of contact.

Note: Tangent is a special case of Secant when the two endpoints of its corresponding chord coincide.

Theorems Related to Circle

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

To prove: OP⟂ XY.

Given: XY is a tangent. 

Construction: Mark a point Q on tangent XY. Join Q to O. That is the centre of the circle. Note that the Q must lie outside the circle. 

Solution: we know by construction that Q is the point outside the circle (if the Q lies inside the circle, the XY will become a secant). Hence, OQ is longer than the radius OP as the radius lies inside. Mathematically, it can be written as, 

OQ＞OP

This will happen for every point on line XY except for the point P, which is the point of contact of the circle on tangent. Hence, OP is the shortest distance of all the distances between point O and the points of line XY. So, OP is Perpendicular to XY. 

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

To Prove: AP = BP

Given: AP and BP are tangents of a circle with centre O. 

Construction: Joint O to P, A, and B. 

Solution:

In APO and BPO

OP = OP (Common)

OA = OB (Radii of the same circle) 

∠OAP = ∠OBP = 90 (the Radius of a circle is perpendicular to the tangent) 

In APO ⩭ BPO (RHS)

AP = BP (CPCT)

3.0Solved Examples

Example 1: If d₁, d₂ (d₂ > d₁ ) are the diameters of two concentric circles and c is the length of a chord of a circle that is tangent to the other circle, prove that d₂² = c² + d₁².

To Prove: d₂² = c² + d₁²

Given: AB = c. 

Construction: Join O to B. 

Solution: It is given that AB is the chord of the bigger circle. 

Hence, OC bisects AB (the line through the centre of the circle bisects the chord of the circle) 

BC = ½ AB = ½ c

Now, in triangle BCO by using Pythagorean theorem. 

BO² = OC² + BC²

12d22=12d12+12c2

d22=d12+c2

or,

d12+c2=d22

Example 2: In Figure, from an external point P, a tangent PT and a line segment PAB are drawn to a circle with centre O. ON is perpendicular to the chord AB. Prove that : (i) PA.PB = PN² – AN² (ii) PN² – AN² = OP² – OT² (iii) PA.PB = PT²

Solution: 

(i) PA can also be written as (PN – AN) similarly, PB = (PN + BN)

PA.PB = (PN – AN) (PN + BN) 

AN = BN (line drawn from the centre of the circle to the chord bisects the cord)

= (PN – AN) (PN + AN) 

(a² – b² = (a+b)(a–b)       

= PN² – AN² 

(ii) As ON⊥PN Using Pythagoras theorem, 

PN² = OP² – ON²

PN² – AN² = (OP² – ON²) – AN² 

= OP² – (ON² + AN²) 

As ON⊥AN

OA² = ON² + AN² 

= OP² – OA² 

OA = OT (radii of the same circle) 

= OP² – OT² 

(iii) From (i) and (ii) 

PA.PB = OP² – OT² 

∠OTP = 90° (radius of the circle makes a right angle with the tangent) hence, 

PT² = OP²– OT² 

= PT² 

Example 3: If a circle touches the side BC of a triangle ABC at P and extended sides AB and AC at Q and R, respectively, prove that AQ = ½(BC + CA + AB)

Solution: 

(The lengths of tangents drawn from an external point to a circle are equal)

BQ = BP 

CP = CR, and 

AQ = AR 

Now, 2AQ = AQ + AQ

2AQ = AQ + AR 

AQ = AB + BQ, and AR = AC + CR

2AQ = (AB + BQ) + (AC + CR) 

2AQ = AB + BP + AC + CP 

2AQ = (BP + CP) + AC + AB 

BC = BP + CP

2AQ = BC + CA + AB 

AQ = ½(BC + CA + AB)

4.0Tips and Tricks

• Remember the Key Property: Always note that the radius (r) is perpendicular to the tangent at the point of contact. This property simplifies many problems.
• Use Diagrams: Draw clear diagrams to visualise the problem and label all known and unknown values for clarity.
• Check Equal Tangents: For tangents drawn from an external point, ensure both tangents are equal in length to cross-check your calculations.
• Practice Proofs: Practice derivations of tangent properties, as they are commonly asked in exams.

5.0Key Features of CBSE Class 10 Maths Notes Chapter 10 Circles

• Concept Clarity: The notes explain tangents, chords, and secants in a clear and organised manner, helping students understand chapter objectives effectively.
• Comprehensive Coverage of Formulas: Key formulas, such as tangent length and radius relationships, are clearly outlined for quick revision.
• Solved Examples with Practical Relevance: Practical examples demonstrate tangent properties in real-world scenarios and prepare students for exams.
• Interactive Practice Problems:  Diverse practice questions encourage independent problem-solving and confidence-building.
• Exam-Friendly Format: Concepts and formulas are presented clearly for efficient last-minute review.

The CBSE Class 10 Maths Notes for Chapter 10 Circles are a priceless tool for understanding the material and performing well on tests because of their characteristics.