Basic Concepts of a Circle

You might have seen many objects that are circular in shape. A circle is defined as the collection of all the

points on a plane that are at equal distance from a given fixed point on the plane. This fixed point is called

centre of the circle and the fixed distance is called the radius.

A line segment joining the centre of a circle with any point on its circumference is called the radius of the circle.

A line that joins two points on the circumference of a circle is called a chord. A chord that passes through the

centre of a circle is called the diameter of the circle. A diameter divides a circle into two equal parts, each is

called a semicircle. Diameter is the longest chord of a circle. The diameter of a circle is twice the radius.

The part of the circumference of a circle between two given points is called an arc.  

A chord divides a circular area into two parts called segments they are major segment and minor segments.

The region between two radii of a circle and any of the arcs between them is called a sector. The diameter of

a circle divides it into two equal segments.

Chords of a Circle

We know that The perpendicular from a point to a line segment is the shortest distance between them. A line

that joins two points on the circumference of a circle is called a chord. A chord passing through the center of

the circle is called diameter. The longest chord of the circle is the diameter.

Theorem: The perpendicular from the centre of a circle to a chord bisects the chord.

Theorem: The line drawn from the centre of a circle to bisect a chord is perpendicular to the

chord. 

Theorem: Equal chords of a circle are equidistant from the centre of the circle.

Theorem: Chords equidistant from the centre of a circle are equal in length.

Let AB be any chord the circle with centre O. Then  is called the angle subtended by the chord at the

centre of the circle. As the chord moves away from the centre, its length and angle subtended by it at the centre

decreases and if it moves closer to the centre its length and angle subtended by it at the centre increases.

Theorem: Chords that subtend equal angles at the centre of a circle are equal in length.

Theorem: Equal chords of a circle subtend equal angles at the centre.

Arc of a Circle

In our daily life we come across many things which are circular in shape. The collection of all the points in a

plane, which are at a fixed distance from a fixed point in the plane, is called a circle. The fixed point is called the

centre of the circle and the fixed distance is called the radius of the circle.

A part of a circle is called an arc. Arcs of a circle that superimpose each other completely are called

congruent arcs. A segment having its endpoints on a circle is called a chord. Diameter is the longest

chord.If two arcs of a circle are congruent, then their corresponding chords are equal and conversely if two

chords of a circle are equal, then their corresponding arcs are congruent.

Theorem: Congruent arcs of a circle subtend equal angles at the centre.

Theorem: The angle subtended by an arc at the centre is double the angle subtended by the arc at any point

on the remaining circle.

Theorem: Angles subtended by an arc at all points within the same segment of the circle are equal.

 

All angles formed in a semi circle are right angles

Cyclic Quadrilaterals

We can draw a circle passing through three non collinear distinct points. The points that lie on a circle are called

concyclic points. So we can say three non collinear points are always concyclic.

Theorem: If a line segment joining two points subtends equal angles at two other points on the same side of the line

segment then all the four points are concyclic.

A quadrilateral whose vertices lie on a circle is called cyclic quadrilateral.

In a cyclic quadrilateral, the sum of opposite angles is always equal to 180o.

If the sum of opposite angles of a quadrilateral is 180o, then the quadrilateral is cyclic.

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