Circles Class 9 Maths Chapter 9 CBSE Notes - 2025-26

Introduction

Circle:

• The locus of the points at a certain distance from a fixed point is defined as a circle.

Chord:

• A chord is a straight line that connects any two points on a circle.

• A chord is represented by the letters $$\text{AB}$$ .

• If the longest chord passes  through the centre of the circle, it is termed as the diameter.

• The radius is twice as long as the diameter. 

• A diameter is referred to as a $$\text{CD}$$.

• A secant is a line that divides a circle in half. 

• $$\text{PQR}$$ is a secant of a circle.

Circumference:

• Circumference refers to the length of a full circle. 

• The circumference of a circle is defined as the border curve (or perimeter) of the circle.

Arc:

• An arc is any section or a part of the circumference.

• A diameter divides a circle into two equal pieces. 

• A minor arc is one that is smaller than a semicircle. 

• A major arc is one that is larger than a semicircle.

• $$\stackrel{⌢}{\text{ADC}}$$ is a minor arc, whereas $$\stackrel{⌢}{\text{ABC}}$$ is a major arc.

Sector:

• A sector is the area between an arc and the two radii that connect the arc's centre and end points.

• A segment is a section of a circle that has been cut off by a chord.

Concentric Circles:

Concentric circles are circles with the same centre.

Theorem $$\text{1}$$:

A straight line drawn from the centre of a circle to bisect a chord which is not a diameter, is at right angles to the chord.

• Given Data:

• Here, $$\text{AB}$$ is a chord of a circle with the centre $$\text{O}$$.

• The midpoint of $$\text{AB}$$ is $$\text{M}$$. 

• $$\text{OM}$$ is joined.

• To Prove:

$$\mathrm{\angle }\text{AMO = }\mathrm{\angle }\text{BMO = 9}{\text{0}}^{\text{0}}$$

• Construction:

Join $$\text{AO}$$ and $$\text{BO}$$.

• Proof:

In $$\mathrm{\Delta }\text{AOM }$$and $$\mathrm{\Delta }\text{BOM }$$

Theorem $$2$$ (Converse of theorem $$1$$):

The perpendicular to a chord from the centre of a circle bisects the chord.

• Given Data:

1. Here, $$\text{AB}$$ is a chord of a circle with the centre $$\text{O}$$.

2. $$\text{OM}\mathrm{\perp }\text{AB}$$

• To Prove:

$$\text{AM = BM}$$

• Construction:

Join $$\text{AO}$$ and $$\text{BO}$$.

• Proof:

In $$\mathrm{\Delta }\text{AOM }$$and $$\mathrm{\Delta }\text{BOM }$$

The transposition of a statement consisting of 'data' and 'to prove' is the converse of a theorem.

We can see how it works by looking at the previous two theorems:

Theorem $$3$$:

Equal chords of a circle are equidistant from the centre.

• Given Data:

• Here, $$\text{AB}$$and $$\text{CD}$$are equal chords of a circle with centre $$\text{O}$$.

• $$\text{OK}\mathrm{\perp }\text{AB}$$ and $$\text{OL}\mathrm{\perp }\text{CD}$$

• To Prove:

$$\text{OK = OL}$$

• Construction:

Join $$\text{AO}$$ and $$\text{CO}$$.

• Proof:

Theorem $$4$$(Converse of theorem $$3$$):

Chords which are equidistant from the centre of a circle are equal.

• Given Data:

• Here, $$\text{AB}$$and $$\text{CD}$$are equal chords of a circle with centre $$\text{O}$$.

• $$\text{OK}\mathrm{\perp }\text{AB}$$ and $$\text{OL}\mathrm{\perp }\text{CD}$$

• $$\text{OK = OL}$$

• To Prove:

$$\text{AB = CD}$$

• Construction:

Join $$\text{AO}$$ and $$\text{CO}$$.

• Proof:

In $$\mathrm{\Delta }\text{AOK}$$and $$\mathrm{\Delta }\text{COL }$$

Theorem $$5$$:

There is one circle, and only one, which passes through three given points not in a straight line.

• Given Data:

Here, $$\text{X, Y}$$and $$\text{Z}$$ are three points not in a straight line.

• To Prove:

A unique circle passes through $$\text{X, Y}$$and $$\text{Z}$$.

• Construction:

• Join $$\text{XY}$$ and $$\text{YZ}$$.

• Draw perpendicular bisectors of $$\text{XY}$$ and $$\text{YZ}$$ to meet at $$\text{O}$$.

• Proof:

Angle Properties (Angle, Cyclic Quadrilaterals and Arcs):

• In figure$$\left(\text{i}\right)$$ , the straight line $$\text{AB}$$students $$\mathrm{\angle }\text{APB}$$ on the circumference.

$$\mathrm{\angle }\text{APB}$$can be said to be subtended by arc $$\text{AMB}$$, on the remaining part of the circumference.

• In fig. $$\left(\text{ii}\right)$$ , arc $$\text{AMB}$$ subtends $$\mathrm{\angle }\text{APB}$$ on the circumference, and it subtends $$\mathrm{\angle }\text{AOB}$$ at the centre.

• In fig. $$\left(\text{iii}\right)$$, $$\mathrm{\angle }\text{APB}$$ and $$\mathrm{\angle }\text{AQB}$$ are in the same segment.

• Now we will go through the theorems based on the angle properties of the circles.

Theorem $$6$$:

The angle at which an arc of a circle subtends at the centre is double the angle which it subtends at any point on the remaining part of the circumference.

• Given Data:

Arc $$\text{AMB}$$subtends $$◻\text{AOB}$$ at the center $$\text{O}$$ of the circle and $$◻\text{APB}$$ on the remaining part of circumference.

• To Prove: 

$$\mathrm{\angle }\text{AOB = 2}\mathrm{\angle }\text{APB}$$

• Construction:

Join $$\text{PO}$$ and produce it to $$\text{Q}$$.

Let, $$\mathrm{\angle }\text{APQ = x}$$ and $$\mathrm{\angle }\text{BPQ = y}$$

• Proof:

Theorem $$7$$:

Angles in the same segment of a circle are equal.

• Given Data:

$$\mathrm{\angle }\text{APB}$$ and $$\mathrm{\angle }\text{AQB}$$ are in the same segment of a circle with center $$\text{O}$$.

• To Prove: 

$$\mathrm{\angle }\text{APB = }\mathrm{\angle }\text{AQB}$$

• Construction:

Join $$\text{AO}$$ and $$\text{BO}$$.

Let arc $$\text{AMB}$$subtend angle $$\text{x}$$ at the center $$\text{O}$$.

• Proof:

Theorem $$8$$:

The angle in a semicircle is a right angle.

• Given Data:

$$\text{AB}$$ is the diameter of a circle with center $$\text{O}$$.

$$\text{P}$$ is any point on the circle.

• To Prove: 

$$\mathrm{\angle }\text{APB = }{90}^{\circ }$$

• Proof:

Cyclic Quadrilaterals:

• If the vertices of a quadrilateral lie on a circle, the quadrilateral is called a cyclic quadrilateral. 

• The vertices are known as concyclic points.

• From the above figure, $$\text{ABCD}$$ is a cyclic quadrilateral. 

The vertices $$\text{A, B, C}$$ and $$\text{D}$$ are concyclic points.

Theorem $$9$$:

The opposite angles of a quadrilateral inscribed in a circle (cyclic) are supplementary.

• Given Data:

$$\text{ABCD}$$ is a cyclic quadrilateral.

 $$\text{O}$$ is the center  of a circle.

• To Prove: 

1. $$\mathrm{\angle }\text{A+}\mathrm{\angle }\text{C = 18}{0}^{\circ }$$

2. $$\mathrm{\angle }\text{B+}\mathrm{\angle }\text{D = 18}{0}^{\circ }$$

• Proof:

Corollary: 

The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

• Given Data:

$$\text{ABCD}$$ is a cyclic quadrilateral.

 $$\text{BC}$$ is produced to $$\text{E}$$

• To Prove: 

$$\mathrm{\angle }\text{DCE}=\mathrm{\angle }\text{A}$$

• Proof:

Alternate Segment Property 

Theorem $$10$$:  

The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

• Given Data:

A straight line $$\text{SAT}$$ touches a given circle with centre $$\text{O}$$ at $$\text{A}$$. $$\text{AC}$$ is a chord through the point of contact $$\text{A}$$.

$$\mathrm{\angle }\text{ADC}$$ is an angle in the alternate segment to $$\mathrm{\angle }\text{CAT}$$ and $$\mathrm{\angle }\text{AEC}$$is an angle in the alternate segment to $$\mathrm{\angle }\text{CAS}$$

• To Prove: 

1. $$\mathrm{\angle }\text{CAT}=\mathrm{\angle }\text{ADC}$$

2. $$\mathrm{\angle }\text{CAS}=\mathrm{\angle }\text{AEC}$$

• Construction:

Draw $$\text{AOB}$$ as diameter and join $$\text{BC}$$ and $$\text{OC}$$.

• Proof:

Theorem $$11$$:  

In equal circles (or in the same circle), if two arcs subtend equal angles at the centres, they are equal.

• Given Data:

$$\text{AXB}$$ and $$\text{CYD}$$ are equal circles with centers $$\text{P}$$ and $$\text{O}$$.

Arcs $$\text{AMD, CND}$$ subtend equal angles $$\text{APB, CQD}$$.

• To Prove: 

$$\text{arc AMD = arc CND}$$

• Proof:

Theorem $$12$$ (Converse of $$11$$):  

In equal circles (or in the same circle) if two arcs are equal, they subtend equal angles at the centres.

• Given Data:

In equal circles $$\text{AXB}$$ and $$\text{CYD}$$, equal arcs $$\text{AMD}$$ and $$\text{CND}$$ subtend $$\mathrm{\angle }\text{APB}$$ and $$\mathrm{\angle }\text{CQD}$$ at the centers $$\text{P}$$ and $$\text{Q}$$ respectively.

• To Prove: 

$$\mathrm{\angle }\text{APB = }\mathrm{\angle }\text{CQD}$$

• Proof:

In case of the same circle:

Figures $$\left(\text{ii}\right)$$ and $$\left(\text{iii}\right)$$ can be considered to be two equal circles which are obtained from figure $$\left(\text{i}\right)$$ and then the above proofs may be applied.

Theorem $$13$$):

In equal circles (or in the same circle), if two chords are equal, they cut off equal arcs.

• Given Data:

In equal circles $$\text{AXB, CYD}$$ with centers $$\text{P}$$ and $$\text{Q}$$ have

 $$\text{chord AB = chord CD}$$

• To Prove: 

$$\text{arc AMB = arc CND}$$

$$\text{arc AXB = arc CYD}$$

• Proof:

In $$\mathrm{\Delta }\text{ABP}$$ and $$\mathrm{\Delta }\text{CDQ}$$

Theorem $$14$$ (Converse of $$13$$):

In equal circles (or in the same circle) if two arcs are equal, the chords of the arcs are equal.

• Given Data:

Equal circles $$\text{AXB, CYD}$$ with centers $$\text{P}$$ and $$\text{Q}$$ have

 $$\text{arc AMB = arc CND}$$

• To Prove: 

$$\text{chord AB = chord CD}$$

• Construction:

Join $$\text{AP, BP, CQ}$$ and $$\text{DQ}$$.

• Proof:

In $$\mathrm{\Delta }\text{ABP}$$ and $$\mathrm{\Delta }\text{CDQ}$$

Theorem $$15$$:

If two chords of a circle intersect internally, then the product of the length of the segments are equal.

• Given Data:

$$\text{AB}$$ and $$\text{CD}$$ are chords of a circle intersecting externally at $$\text{P}$$.

• To Prove: 

$$\text{AP }\times \text{ BP = CP}\times \text{DP}$$

• Construction:

Join $$\text{AC}$$ and $$\text{BD}$$.

• Proof:

In $$\mathrm{\Delta }\text{APC}$$ and $$\mathrm{\Delta }\text{DPB}$$

Theorem $$16$$:

If two chords of a circle intersect externally, then the product of the lengths of the segments are equal.

• Given Data:

$$\text{AB}$$ and $$\text{CD}$$ are chords of a circle intersecting externally at $$\text{P}$$.

• To Prove: 

$$\text{AP }\times \text{ BP = CP}\times \text{DP}$$

• Construction:

Join $$\text{AC}$$ and $$\text{BD}$$.

• Proof:

In $$\mathrm{\Delta }\text{ACP}$$ and $$\mathrm{\Delta }\text{DBP}$$

Theorem $$17$$:

If a chord and a tangent intersect externally, then the product of the lengths of the segments of the chord is equal to the square on the length of the tangent from the point of contact to the point of intersection.

• Given Data:

A chord $$\text{AB}$$ and a tangent $$\text{TP}$$ at a point $$\text{T}$$ on the circle intersect at $$\text{P}$$.

• To Prove: 

$$\text{AP }\times \text{ BP = P}{\text{T}}^{\text{2}}$$

• Construction:

Join $$\text{AT}$$ and $$\text{BT}$$.

• Proof:

Test for Concyclic Points:

1. Conversely, the statement, 'Angles in the same segment of a circle are equal', is one test for concyclic points. 

We state:

If two equal angles are on the same side of a line and are subtended by it, then the four points are concyclic. 

In the figure, if $\mathrm{\angle }P=\mathrm{\angle }Q$and the points $P,Q$ are on the same side of $AB$, then the points $A,B,P$ and $Q$are concyclic.

2. Converse of 'opposite angles of a cyclic quadrilateral are supplementary' is one more test for concyclic points.

We state:

If the opposite angles of a quadrilateral are supplementary, then its vertices are concyclic. 

In the figure, if $\mathrm{\angle }\text{A}+\mathrm{\angle }C={180}^{\circ }$ then $\text{A, B, C}$ and $\text{D}$ are concyclic points.

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Conclusion 

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