CIRCLES

CIRCLES

1. A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any point of the circle and the centre is called the radius.

2. Theorems

   • Equal chords of a circle subtend equal angles at the centre .
   • If the angles subtended by the chords of a circle at the centre are equal , then the chords are equal.
   • The perpendicular from the center of a circle to a chord bisects the chord .
   • The line drawn through the center of the circle to bisect a chord is perpendicular to the chord .
   • The is one and only one circle passing through three given non-collinear points .
   • Equal chords of a circle ( or of congruent circles ) are equidistant from the centre ( or centres ) .

3. Theorems

   • Chords equidistant from the centre are equal in length .
   • 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 .
   • Angles in the same segment of a circle are equal .
   • If a line segment joining two points subtends equal angles at two other points lying on the same slide of the line containig the line segment , the four points lie on a circle ( i.e. they are concyclic ) .
   • If the the sum of a pair of opposite angles of a quadrilateral is 180 degrees .
   • If the sum of a pair of opposite angles of a quadrilateral is 180 degrees , the quadrilateral is cyclic .

4. PARTS OF A CIRCLE

   • DIAMETER : It is any line segment that passes through the centre of the circle and its end points lie on the circle .
   • RADIUS : It is any line segment from the circles center to the perimeter of the circle .
   • CHORD : A chord is a straight line segment whose both endpoints lie on the perimeter of the circle .
   • SEGMENT : A segment is a region bounded by a chord of a circle and the intercepted arc of the circle . The smaller area is known as Minor segment and the larger is known as Major segment .
   • SECTOR ; A sector is the portion of the circle enclosed by two radii and an arc . The smaller area is known as Minor sector and the larger area is known as Major sector .