CIRCLES

CIRCLES

1. DEFINITION

   • A CIRCLE is defined as the locus of all points that are equidistant from a given point. In the given figure, O is the fixed point.

2. TERMS RELATED TO CIRCLES

   • CENTRE
     • The fixed point in the interior of the circle which is equidistant from all the other points on the circumference
   • RADIUS
     • The distance between the centre and any point on the circumference
   • DIAMETER
     • The line passing through the centre, which joins any two points on the circumference
   • CIRCUMFERENCE
     • The path traced by all the points, which are at a fixed distance from a given point
   • CHORD
     • The line joining any two points on the circumference
   • ARC
     • MINOR
     • MAJOR
     • A chord divides the circumference of the circle into two parts. The shorter one is called the Minor Arc. The longer one is called Major Arc.
   • SEGMENT
     • MINOR
     • MAJOR
     • The area between a chord and an arc is called a Segment. If the corresponding arc is minor, then it is called Minor Segment. If the corresponding arc is major, then the segment is called Major Segment.
   • SECTOR
     • MINOR
     • MAJOR
     • The area between two radii and any one of the arc is called a Sector. If the corresponding arc is minor, then it is called Minor Sector. If the corresponding arc is major, then the segment is called Major Sector.
   • SECANT
     • If we extent a chord on both sides, then that figure is called a Secant.
   • TANGENT
     • It is a line that just touches the circumference of the circle at one point. It is perpendicular to the radius.

3. IMPORTANT THEOREMS

   • 1. There is one and only one circle passing through three given non-colliner points. In the given figure only one circle passes through the non-collinear P, Q and R.
   • 2. 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 circumference. In the given figure, ∠POQ = 2∠PAQ
   • 3. Angles in the same segment of a circle are equal. In the given figure, ∠PAQ = ∠PBQ
   • 4. 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 are concyclic. In the given figure, A, B, C and D are concyclic.
   • 5. The sum of either pair of opposite angles of a cyclic quadrilateral is 180°. In the given figure, ∠A + ∠C = ∠B + ∠D = 180°.

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