## Revision Notes ofÂ 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, â€˜Oâ€™ 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.Â - 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:Â**

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

(ii) Diameter = 2 Ã— 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:Â**

(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__.Â - The part containing the
__minor arc__is called the__minor segment__, and the part containing the__major arc__is called the__major segmen__t.Â

- 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 1Â Â**

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

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

__To prove:__âˆ AOB = âˆ COD

__Proof:__Â

In â–³AOB and â–³COD,Â

AO = CO (radii of the same circle)

BO = DO (radii of the same circle)

AB = CD (given)Â

âˆ´ â–³AOB â‰… â–³COD (SSS)Â

Hence, âˆ AOB = âˆ 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 areÂ equal.

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

__To prove:__PQ = RS

__Proof:__In â–³POQ and â–³ROS,Â

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

and âˆ POQ = âˆ ROS (given)

âˆ´ â–³POQ â‰… â–³ROS (SAS)

âˆ´ PQ = RS. (c.p.c.t.)Â

**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 âŠ¥ AB.Â

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

__Proof:__Â

In â–³ODA and â–³ODB,Â

OA = OB (radii of the same circle)

OD = OD (common)Â

âˆ ODA = âˆ ODB (each is a rt. angle)

â–³ODA â‰… â–³ODB (R.H.S.)

AD = DB (c.p.c.t.)Â

**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.Â

__To prove:__OL âŠ¥ PQ.Â Construction: Joint OP and OQ.

__Proof:__

In â–³OLP and â–³OLQ,

OP = OQ (radii of the same circle)

PL = QL (given)

OL = OL (common)

âˆ´ â–³OLP â‰… â–³OLQ (SSS)

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

âˆ´ âˆ OLP = âˆ OLQ = 90Â°

Hence, OL âŠ¥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.Â 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 â–³AOM and â–³CON,

OA = OC (radii of the same circle)

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

âˆ OMA = âˆ ONC = 90Â°

âˆ´ â–³AOM â‰… â–³CON (R.H.S.)

âˆ´ 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:__Â Â

OM âŠ¥ AB â‡’ 1/2 AB = AMÂ

ON âŠ¥ CD â‡’ 1/2 CD = CN

Consider â–³AOM and â–³CON,Â

OA = OC (radii of the same circle)

OM = ON (given)Â

âˆ OMA = âˆ ONC = 90Â° (given)

â–³AOM â‰… â–³CON (RHS congruency)

AM = CN â‡’ 1/2 AB = 1/2 CD â‡’ AB = CD

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

**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.Â

__Given:__O is the centre of the circle.

__To prove:__âˆ BOC = 2âˆ BAC

__Construction:__Join O to A.

__Proof:__

In â–³AOB,

OA = OB (radii of the same circle)Â

â‡’ âˆ 1 = âˆ 2

Similarly in â–³AOC,

âˆ 3 = âˆ 4

Now, by exterior angle property,

âˆ 5 = âˆ 1 + âˆ 2

âˆ 6 = âˆ 3 + âˆ 4

â‡’ âˆ 5 + âˆ 6 = âˆ 1 + âˆ 2 + âˆ 3 + âˆ 4

â‡’ âˆ 5 + âˆ 6 = 2âˆ 2 + 2âˆ 3Â = 2(âˆ 2 + âˆ 3)

â‡’ âˆ BOC = 2âˆ BAC

**Theorem 9**

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

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

__To prove:__âˆ ACB = âˆ ADB

__Construction:__Join OA and OB.

__Proof:__Â

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

Hence, âˆ AOB = 2âˆ ACBÂ

âˆ AOB = 2âˆ ADB

So, âˆ ACB = âˆ ADB

In fig. (ii), we have,Â

Reflex âˆ AOB = 2âˆ ACBÂ and Reflex âˆ AOB = 2âˆ ADB

2âˆ ACB = 2âˆ ADB

âˆ´ âˆ ACB = âˆ ADB

**Theorem 10**

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

**Theorem 11**

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

__Given:__Let ABCD be a cyclic quadrilateral

__To prove:__âˆ A + âˆ C = 180Â° and âˆ B + âˆ D = 180Â°

__Construction:__Join OB and OD.

__Proof:__Â

âˆ BOD = 2âˆ BADÂ

âˆ BAD = 1/2 âˆ BOD

Similarly, âˆ BCD = 1/2 reflex âˆ BODÂ

âˆ´ âˆ BAD + âˆ BCD = 1/2 âˆ BOD + 1/2 reflex âˆ BOD

= 1/2 (âˆ BOD + reflex âˆ BOD) = 1/2 Ã—360Â°

âˆ´ âˆ A + âˆ C = 180Â°

Similarly, âˆ B + âˆ D = 180Â°

**Theorem 12**

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