Circles

Introduction: A circle is a set of all those points in a plane which are equidistant from a given point, called the centre of the circle, in the same plane. The distance of any point on the circle from the centre is known as the radius of the circle.Remember:• A circle has one and only one centre.• A circle has infinite number if radii and diameter.

Terminology: In the given figure, point O is the centre

and OA is the radius of the circle. Generally, radius is denoted by ‘r’.

E

n

X

mY

P

Q

Ar

S

ɵ

O

Chord: A line segment joining two distinct points on the circle is

known as a chord. for example, in the above figure, XY is a chord.

Diameter: The longest chord that can be drawn in a circle is known

as the diameter, generally denoted by ‘d’. It always passes through the centre of the circle.

For example, in the given figure, PQ is a diameter Length of the diameter is double that of the radius. Diameter,

d = 2rTangent: A line which lies in the plane of a circle and touches the

circle at only one point is known as a tangent, i.e. there is only one point common to a circle and the tangent. The point at which the tangent touches the circle is called the point of contact or point of tangency.

For example, in the given figure, line m is the tangent. E is the point of contact.

Secant: A line which passes through two distinct points

on a circle is known as a secant. For example, in the given figure, line n is the secant.Arc: A secant (or a chord) divides the circle in two

parts, each of the two parts along with the two points common with the secant (or a chord) is known as an arc. The smaller one is the minor arc and the larger one is the major arc. An arc is given a three letter name.

For example, in the given figure, arc XEY is the minor and arc XSY is the major arc.

Central Angle: An angle made by two points on the circle at the

centre is known as a central angle.

For example, in the given figure, angle AOQ is the central angle made by arc ASQ. If we draw AQ then we can also say “angle AOQ is the central angle of the chord AQ”

Remember:- Angular measure (or simply measure) of an arc ASQ is denoted by m(arc ASQ). Also, m(arc ASQ) = m angle AOQ = ɵ.- Angular measure of a minor arc is always less than 1800. - Angular measure of the major arc = 360o – the angular measure of the corresponding minor arc.- Angular measure of a semicircle is 1800.- Angular measure of a circle is 360o.