Basic Property Of Circle (Important)

3. Diameter: Chord passing through the centre of the circle. It is the largest chord of the circle. Diameter is twice the radius of a circle. When P and Q are ends of diameter, then both arcs are semi-circle.

4. Segment: The region between a chord and either of its arcs is called a segment of the circular region or simply a segment of the circle.

5. Sector: The region between an arc and the two radii, joining the centre to the end points of the arc is called a sector.

6. Properties:

i) Equal chords of a circle subtend equal angles at the centre. Reverse of it also holds true, i.e, if the angles subtended by the chords of a circle at the centre are equal, then the chords are equal.

ii) Equal chords of a circle (or of congruent circles) are equidistant from the centre (or centres). Converse is also true, i.e., Chords equidistant from the centre of a circle are equal in length.

iii) The perpendicular from the centre of a circle to a chord bisects the chord.

iv) There is one and only one circle passing through three given non-collinear points.

v) The length of the perpendicular from a point to a line is the distance of the line from the point.

vi) 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.

vii) Angles in the same segment of a circle are equal and converse is also true, i.e., 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 lie on a circle

viii) Cyclic quadrilateral is a quadrilateral having all of its vertices on the circle.

ix) The sum of either pair of opposite angles of a cyclic quadrilateral is 180º. Converse is also true, i.e., if the sum of a pair of opposite angles of a quadrilateral is 180º, the quadrilateral is cyclic.