Circles Class 9

 Circles 

Circle 

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.

Centre of the circle / Radius

The fixed point is called the centre of the circle and the fixed distance is called the radius of the circle. In Fig, O is the centre and the length OP is the radius of the circle.

Parts of the circle 

A circle divides the plane on which it lies into three parts. They are: (i) inside the circle, which is also called the interior of the circle; (ii) the circle and (iii) outside the circle, which is also called the exterior of the circle (see Fig). The circle and its interior make up the circular region.

Cord / Diameter 

The line segment PQ is called a chord of the circle (see Fig). The chord, which passes through the centre of the circle, is called the diameter of the circle.

The diameter is the longest chord and all diameters have the same length, which is equal to two times the radius.

Arc 

A piece of a circle between two points is called an arc.

The longer one is called the major arc PQ and the shorter one is called the minor arc PQ. The minor arc PQ is also denoted by  

 and the major arc PQ by  

 where R is some point on the arc between P and Q.

       

Semicircle 

When P and Q are ends of a diameter, then both arcs are equal and each is called a semicircle.

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.

Two types of segments 

1. major segment 
2. minor segment

Sector

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

Two types of sectors 

1. minor sector
2. major sector 

The region OPQ is the minor sector and the remaining part of the circular region is the major sector.

When two arcs are equal, that is, each is a semicircle, then both segments and both sectors become the same and each is known as a semicircular region.

Angle Subtended by a Chord at a Point

`angle POQ` is the angle subtended by the chord PQ at the centre O, `anglePRQ` and  `anglePSQ` are respectively the angles subtended by PQ at points R and S on the major and minor arcs PQ.

Relationship between the size of the chord and the angle subtended by it at the centre

Longer is the chord, the bigger will be the angle subtended by it at the centre. What will happen if you take two equal chords of a circle? Equal chords of a circle subtend equal angles at the centre.

Theorem 1: Equal chords of a circle subtend equal angles at the centre.

To Prove:  `angleAOB =  angleCOD`

In `triangleAOB` and `triangleCOD`

OA = OC (Radii of a circle)

OB = OD (Radii of a circle)

AB = CD (Given)

`therefore triangle AOB ≅ triangleCOD` (SSS rule)

This gives  `angleAOB =  angleCOD`  (by CPCT)

Theorems 👉 Click here