## Revision Notes of Chapter 10 Circles Class 9th Math

**Topics in the Chapter**

- Terms related to Circles
- Fundamentals of Circles
- Theorems

**Terms Related to Circles**

- The locus of a point which moves in a plane in such a manner that its distance from a given fixed point is always constant, is called a
__circle__. - The fixed point is called the
__centre__and constant distance is called the__radius__of the circle.

In the figure, ‘O’ is centre and OP = r is a radius. We denote it by C(O, r).

- A line segment, terminating (or having its end points) on the circle is called a
__chord__. A chord, passing through the centre is called a__diameter__of the circle. - A line which intersects a circle in two distinct points is called a
__secant__of the circle. - A line intersecting the circle in exactly one point is called a
__tangent__to the circle.

- In the figure, PQ is a chord, AB is a diameter, XY is a secant and ST is a tangent to the circle at C.

**Note:**

(i) Diameter is the longest chord in a circle.

(ii) Diameter = 2 × Radius

- The length of the complete circle is called its circumference, whereas a piece of a circle between two points is called an
__arc__.

**Note:**

(i) A diameter of a circle divides it into two equal arcs, each of which is called a

__semicircle__.

(ii) If the length of an arc is less than the semicircle, then it is a minor arc, otherwise, it is a

__major arc__.

- The region consisting of all points lying on the circumference of a circle and inside it is called the interior of the circle.
- The region consisting of all points lying outside a circle is called the exterior of the circle.
- The region consisting of all points which are either on the circle or lie inside the circle is called the circular region.

- A chord of a circle divides it into two parts. Each part is called a
__segment__. - The part containing the
__minor arc__is called the__minor segment__, and the part containing the__major arc__is called the__major segmen__t.

- A quadrilateral of which all the four vertices lie on a circle is called a cyclic quadrilateral. The four vertices A, B, C and D are said to be concyclic points.

**Fundamentals of Circles**

- Equal chords of a circle (or of congruent circles) subtend equal angles at the centre.
- If two chords of a circle subtend equal angles at the centre, then the chords are equal.
- The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
- The perpendicular from the centre of a circle to a chord bisects the chord.
- Equal chords of a circle are equidistant from the centre whereas the equidistant chords from the centre are equal.
- Chords corresponding to equal arcs are equal.
- Congruent arcs of a circle subtend equal angles at the centre.
- The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part.
- Angles in the same segment are equal, whereas the angle in a semicircle is a right angle.
- The sum of either pair of opposite angles of a cyclic quadrilateral is 180Âº.
- If the opposite angles of a quadrilateral are supplementary, then the quadrilateral is cyclic.
- If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, then the four points are cyclic.

**Theorems**

**Theorem 1**

__Statement:__Equal chords of a circle subtend equal angles at the centre.

__Given:__AB and CD are chords of a circle with centre O, such that AB = CD.

__To prove:__∠AOB = ∠COD

__Proof:__

In △AOB and △COD,

AO = CO (radii of the same circle)

BO = DO (radii of the same circle)

AB = CD (given)

∴ △AOB ≅ △COD (SSS)

Hence, ∠AOB = ∠COD (c.p.c.t.)

**Theorem 2**

__Statement:__If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal.

__Given:__Two chords PQ and RS of a circle C(O, r), such that ∠POQ = ∠ROS.

__To prove:__PQ = RS

__Proof:__In △POQ and △ROS,

OP = OQ = OR = OS = r (radii of the same circle)

and ∠POQ = ∠ROS (given)

∴ △POQ ≅ △ROS (SAS)

∴ PQ = RS. (c.p.c.t.)

**Theorem 3**

__Statement:__The perpendicular from the centre of a circle to a chord bisects the chord.

__Given:__AB is the chord of a circle with centre O and OD ⊥ AB.

__To prove:__AD = DB Construction: Join OA and OB.

__Proof:__

In △ODA and △ODB,

OA = OB (radii of the same circle)

OD = OD (common)

∠ODA = ∠ODB (each is a rt. angle)

△ODA ≅ △ODB (R.H.S.)

AD = DB (c.p.c.t.)

**Theorem 4**

__Statement:__The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.

__Given:__A chord PQ of a circle C(O, r) and L is the mid-point of PQ.

__To prove:__OL ⊥ PQ. Construction: Joint OP and OQ.

__Proof:__

In △OLP and △OLQ,

OP = OQ (radii of the same circle)

PL = QL (given)

OL = OL (common)

∴ △OLP ≅ △OLQ (SSS)

Also, ∠OLP + ∠OLQ = 180° (linear pair)

∴ ∠OLP = ∠OLQ = 90°

Hence, OL ⊥PQ

**Theorem 5**

__Statement:__There is one and only one circle passing through three given non-collinear points.

**Theorem 6**

__Statement:__Equal chords of a circle (or of congruent circles) are equidistant from the centre (or centres).

__Given:__AB and CD are two equal chords of a circle. OM and ON are perpendiculars from the centre to the chords AB and CD.

__To prove:__OM = ON.

__Construction:__Join OA and OC.

__Proof:__

In △AOM and △CON,

OA = OC (radii of the same circle)

MA = CN (Since, OM and ON are perpendicular to the chords and it bisects the chord and AM = MB, CN = ND)

∠OMA = ∠ONC = 90°

∴ △AOM ≅ △CON (R.H.S.)

∴ OM = ON (c.p.c.t.)

Equal chords of a circle are equidistant from the centre.

**Theorem 7**

__Statement:__Chords equidistant from the centre of a circle are equal in length.

__Given:__OM and ON are perpendiculars from the centre to the chords AB and CD and OM = ON.

__To prove:__Chord AB = Chord CD.

__Construction:__Join OA and OC.

__Proof:__

OM ⊥ AB ⇒ 1/2 AB = AM

ON ⊥ CD ⇒ 1/2 CD = CN

Consider △AOM and △CON,

OA = OC (radii of the same circle)

OM = ON (given)

∠OMA = ∠ONC = 90° (given)

△AOM ≅ △CON (RHS congruency)

AM = CN ⇒ 1/2 AB = 1/2 CD ⇒ AB = CD

The two chords are equal if they are equidistant from the centre.

**Theorem 8**

__Statement:__The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

__Given:__O is the centre of the circle.

__To prove:__∠BOC = 2∠BAC

__Construction:__Join O to A.

__Proof:__

In △AOB,

OA = OB (radii of the same circle)

⇒ ∠1 = ∠2

Similarly in △AOC,

∠3 = ∠4

Now, by exterior angle property,

∠5 = ∠1 + ∠2

∠6 = ∠3 + ∠4

⇒ ∠5 + ∠6 = ∠1 + ∠2 + ∠3 + ∠4

⇒ ∠5 + ∠6 = 2∠2 + 2∠3 = 2(∠2 + ∠3)

⇒ ∠BOC = 2∠BAC

**Theorem 9**

__Statement:__Angles in the same segment of a circle are equal.

__Given:__Two angles ∠ACB and ∠ADB are in the same segment of a circle C(O, r).

__To prove:__∠ACB = ∠ADB

__Construction:__Join OA and OB.

__Proof:__

In fig. (i), we know that, angle subtended by an arc of a circle at the centre is double the angle subtended by the arc in the alternate segment.

Hence, ∠AOB = 2∠ACB

∠AOB = 2∠ADB

So, ∠ACB = ∠ADB

In fig. (ii), we have,

Reflex ∠AOB = 2∠ACB and Reflex ∠AOB = 2∠ADB

2∠ACB = 2∠ADB

∴ ∠ACB = ∠ADB

**Theorem 10**

__Statement:__If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment, the four points lie on a circle (i.e. they are concyclic).

**Theorem 11**

__Statement:__The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.

__Given:__Let ABCD be a cyclic quadrilateral

__To prove:__∠A + ∠C = 180° and ∠B + ∠D = 180°

__Construction:__Join OB and OD.

__Proof:__

∠BOD = 2∠BAD

∠BAD = 1/2 ∠BOD

Similarly, ∠BCD = 1/2 reflex ∠BOD

∴ ∠BAD + ∠BCD = 1/2 ∠BOD + 1/2 reflex ∠BOD

= 1/2 (∠BOD + reflex ∠BOD) = 1/2 ×360°

∴ ∠A + ∠C = 180°

Similarly, ∠B + ∠D = 180°

**Theorem 12**

__Statement:__If the sum of a pair of opposite angles of a quadrilateral is 180°, the quadrilateral is cyclic.