Circle is the locus of all such points which are equidistant from a fixed point, this point is known as centre while distance of any point from centre defined as radius of circle.
Here O is fixed point called centre of the circle. r is called the radius of the circle.
Remember:
• A line joining two points on a circle in known as the chord of the circle.
• A line which intersect a circle in two distinct points is called a secant of the circle .
• Any line segment PQ passing through the centre O of the circle and having its two end points P and Q on the circle is called a diameter.
Note:
Length of a diameter is twice the length of the radius of the circle.
• Major and Minor arc: An arc whose length is less than the semicircular arc is called minor arc and if greater than semicircular arc is called major arc.
Segment of circle: The region enclosed by an arc and its corresponding chord is called a segment of the circle. The segment containing the minor arc is called the minor segment and the segment containing the major arc is called the major segment of the circle.
Remember: The minor and major segments of a circle are called the alternate segments of the circle.
Sector of a circle: The region enclosed by an arc of a circle and its two bounding radii is called a sector of the circle as shown in given figure.
• Concentric circles: Circles which have the same centre and different radii are called concentric circles. In figure C(O, r) and C(O, R) are concentric circles having the same centre O but different radii r, R respectively.
Semicircular region: When two arcs are equal that is each is semicircle, then both segments and both sectors become the same and each is known as semicircular region as shown in adjoining figure.
Quadrant of a circle: One-fourth of a circle is called a quadrant. Thus, in the adjoining figure, OBCO is a quadrant of the circle C(O, r).
Congruent circles: The circles C(O, r) and C(O′, s) are said to be congruent only when r = s.
Cyclic quadrilateral: A quadrilateral ABCD is said to be cyclic if all its vertices lie on a circle. Points lying on a circle are said to be concyclic.
In the adjoining figure, ABCD is cyclic quadrilateral and hence the points A, B, C, D are concyclic.
Remember: Of any two chords of a circle, the larger chord is nearer to the centre.
• The angle subtended by an arc of a circle 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.
• The angle in a semicircle is a right angle.
• If all vertices of a quadrilateral lie on a circle, it is called a cyclic quadrilateral.
• The opposite angles of a cyclic quadrilateral are supplementary.
• If the sum of any pair of opposite angles of quadrilateral is 180°, then it is cyclic.
• If one side of a cyclic quadrilateral is produced, then the exterior angle is equal to the interior opposite angle.
• An isosceles trapezium is cyclic.