☼ Arc of a circle and its length  :

• Arc of a circle :  If  A and B are any two distinct points on a circle, then the set of all the points of the circle lying in any closed half-plane of  is called an arc of a circle.

The arc AB is denoted by .

An arc of a circle can be minor arc, major arc or semicircular arc.

▪ Minor arc  :  If  A and B are the points of a circle with centre O, then the set of the points on the circle lying in the interior of A OB, together with the points A and B, is called the minor  .

▪ Major arc  :  If  A and B are the points of a circle with centre O, then the set of the points on the circle lying in the exterior of  AOB, together with the points A and B is called the major .

▪ Semicircular arc  :  If  A and B are the end–points of a diameter of a circle with center O, then the set of the points on the circle lying in any half – plane of a circle, together with the points A and B is called the semicircular arc .

If   is a chord of the circle with centre O, other than a diameter, then AOB is called the angle subtended by the minor  at the center.

▪ Minor arc  :  If  A and B are the end–points of an arc of a circle and  if  the  points  of    other  than  the  end–points  and  the  center  of  the  circle lie in the different half–planes of  , then is a minor arc.

Length  of   minor  arc  ( l )   = 

▪ Major arc  :  If  A and B are the end–points of an arc of a circle and  if  the  points  of  other  than  the  end–points  and  the  center  of  the  circle lie in the same half–planes of  , then is a major arc.

Length  of   major arc  ( l )   = – 

▪ Semicircular arc  :  If  A and B are the end–points of an arc of a circle and  if    passes through the center of the circle, then  is a semicircular arc.

Length  of  semicircular  arc  ( l )   =

● Congruent arcs  :  If  the lengths of  two arcs of  the same circle  are equal, then they are called  congruent arcs.

If    and  are congruent, then in notations it is denoted as  .

The angles subtended by two congruent minor arcs at the centre are congruent.

 If  two arcs of the same circle are congruent, then the chords of  the circle corresponding to them are also congruent.

  The measure of the angle, subtended by a minor arc of a circle at the centre, is twice the measure of the angle subtended by that arc at any point on the remaining part of the circle.

i.e.  m ∠ APB  =   2 ×  m ∠ ACB

An angle inscribed in a semicircle is a right angle.

↦ An angle inscribed in a semicircle is a right angle.

i.e.  m ∠ ACB  =   90⁰

● Segment of circle  :  If    is an arc of a circle, then     is called a segment of  the circle. is also called the arc of a segment of  the circle.

↦ Segments of circle are called respectively minor segment of the circle or major segment of the circle or semicircular segment of the circle according as its arc is a minor arc or major arc or a semicircle.

● Angle in a segment of a circle  :  If  a segment of  a  circle is ,  then  any  angle inscribed in   is called an angle in the segment of the circle.

↦Angles in the same segment of a circle are congruent.

i.e.  m ∠ACB  =  m ∠ ADB  =  m ∠AEB

↦ If a line-segment, joining two points subtend congruent angles at two distinct points lying in the same half-plane of the line containing this line-segment, then all these four points are on the same segment of a circle ( concyclic ).

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