Class 10 Mathematics Note

Circle

Circle

A circle close curve of locus of set of all points that are equidistant from a given point called the centre. The distance between any of the points and the centre is called the radius.

In the given figure, `O’ is the centre and `r’ is the radius of the circle.

 

Semi-circle

Half the circle separated by the diameter on the either side is called semi – circle. The full arc of a semicircle always measures 180°. 

 

Chord and the Diameter

The straight line joining any two points on the circumference of circle is called chord. The chord which passes through the centre of the circle is called diameter. Diameter if the longest chord of the given circle. The chord divides the circle into major segment and minor segments.

 

Arc:

 

Part of the circumference cut off by the chord is called an arc. AQB and APB is an arc separated by the line segment AB.

 

Sector

A circular sector or circle sector is the portion of a disk enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector. In the diagram, θ is the central angle in radians, the radius of the circle, and is the arc length of the minor sector.

 

 

Secant

A straight line that passes through the two points of the circumference of circle is called secant. In the given figure, CD is the secant. 

 

Tangent:

A tangent like is perpendicular to the radius drawn to the point of contact .If a line is perpendicular to a radius at its outer endpoint, and then it is tangent to the circle.

 

 

Inscribed angle

 An angle between the two chords which have a common endpoint is called inscribed angle. This common endpoint forms the vertex of the inscribed angle. The other two endpoints define what we call an intercepted arc on the circle.∠ ABC is an inscribed angle and AC is a the corresponding arc  .

 

Centre angle

An angle formed by the radius of the circle is called centre angle. .∠ AOC is an centre angle  and  AC is a the corresponding arc.

 

Common chord

If the two or more than two circles intersect, some chords can be common called common chords.AC is the common chords.

 

Theorems

Centre angle is double of the inscribed angle standing on the same arc.

 

Given: ∠ QPR and ∠ QSR   are standing on the same arc QR

To prove: ∠ QSR =2 ∠ QPR

S.NO

 Statement

Reason  

1

2.

 

3.

4.

5.

6.

 In ΔQSP  ,QS  =  QP  ∠ PQS =∠ SPQ

∠ PQS + ∠ SPQ = ∠ QST

 

2∠ SPQ  =  ∠ QST

Like wise,2∠ SPR  =  ∠RST 

2 (∠ SPQ +∠ SPR  )  =  ∠ QST +∠RST 

2 ∠ QPR  =∠ QSR

Radii of the same circle

Exterior angle is equal to the sum of two opposite interior angle.

From 2

Same as above

Adding 3 and 4

From 5

 

 

 

Prove that opposite angles of the cyclic quadrilateral is supplementary.

 

 Given: ABCD is a quadrilateral

To prove:  ∠ B + ∠ D = 180°, ∠ A  + ∠ C  = 180°

     

S.No

 Statement

Reason

1.

2.

3.

4.

5.

∠AOC   =  2∠ADC 

Reflex angle ∠AOC    =  2∠ABC 

360°  =  2(∠ADC   +∠ABC  )

180°  = ∠D  +∠B 

∠ A  + ∠ C  = 180°

 Angle at centre is double of inscribed angles

Angle at centre is double of inscribed angles

 Adding  1 and 2

From 3

By similar method

 

 

Examples 1

In the given figure AB and EF are parallel to each other . Prove that CDEF is a cyclic quadrilateral.

Soln

                                                                                                                                                        Given: AB || EF, ABCD is the cyclic quadrilateral

To prove: CDEF is the cyclic quadrilateral

 

S.No

 Statement

 Reason

1.

2.

 

3.

 

∠ ABC  = ∠ CDC, ∠ DAB =   ∠ DCF

∠ ABC  +∠ DCF =  180°

∠ DAB  +∠ DEF =  180°

∠ DCF  +∠ DEF =  180°

∠CDE  +∠ EFC  =  180°

In  cyclic quadrilateral exterior angle formed is equal to the opposite interior angle

 Sum of the co- interior angles.

 

 From 1 and  2

 

 

Example 2

 

Soln

Given: NPS, MAN and RMS are the straight line

To prove:  PQRS is a cyclic quadrilateral

Construction: join AQ

S.NO

Statemen

Reason

1.

2.

3.

4.

5.

6.

7.

 

 

∠ PNA = ∠PQA

∠ AQR  = ∠ AMS

∠ AQR  + ∠ AMR  = 180°

∠ AMR  = ∠ PNA +∠ PSR

∠ AMR  = ∠PQA +∠ PSR

∠ AQR+∠PQA +∠ PSR  = 180°

∠PQR + ∠ PSR    = 180°

 

 

Standing on the same arc AQ

Exterior angle of the cyclic quadrilateral is equal to the opposite interior angle

Opposite angle of the cyclic quadrilateral is supplementary

Ext angle of the triangle is equal to the sum of two  opp. interior angles

From 1

From 5

From 6

 

 

 

Example 3

O is the centre of the circle .PS || OR   and PQ in the diameter. Prove that arc  SR  =arc  RQ

 

Given:  O is the centre of the circle .PS || OR   and PQ in the diameter

 To prove: arc SR =  arc RQ

Statement

Reasons

1. ∠  ROQ  = arc RQ 

2.∠ SPQ  =  1/2   arc SQ 

3.∠ ROQ  = ∠ SPQ 

4. RQ  =  1/2  arcSQ

5.arc  SR  =arc  RQ

 

Relation between central angle and its opposite arc

 Circumference angle and its opposite arc

PS||QR  and corresponding angles

From 1 ,2 and 3

From statement 4

 


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