**Notes of chapter: Circles are presented below. Indepth notes along with worksheets and NCERT Solutions of class 9.**

**(1) Circle-**

A **circle **is the collection of all points in a plane, which are equidistant from a fixed point in the plane.

Figure of circle is showing below-

O is a fixed point.

**Parts of the circle-**

**(i) Center-**

The fixed point of circle is known as **center**.

**Eg:-** O is center of the circle.

**(ii)Radius-**

The distant from center of the circle to the boundary of circle is known as **radius.**

**Eg:-** OP is distance from center and boundary of the circle.

**(iii) Chord-**

The line segments which touches boundary of the circle is known as **chord.**

**Eg:- **AB is a chord of the circle.

**(a) Diameter-**

The chord which passes from the center is known as **diameter.**

**Eg:-** CD is a diameter of the circle.

**Properties of the diameter-**

**(A)** It is longest chord of the circle.

**(B)** All diameters of a circle are equal in lengths.

**(C)** Diameter is twice of the radius of the circle.

Diameter = 2radius

**(iv)**The part of a circle between two points is called an **arc.**

**Eg:-** In figure below, points P and Q divides circle in two parts which are known as arcs. One arc is PAQ and second arc is PBQ.

**Types of arcs-**

**(a) Major arc –**

The longer part of a circle between two points on a circle is known as **major arc.**

**Eg:- **In figure below, PRQ is a major arc.

**(b) Minor arc-**

The shorter part of a circle between two points on a circle is known as **minor arc.**

**Eg:-** In figure below PQ is a minor arc.

**(c)Semicircle-**

When P and Q are ends of a diameter then both arcs are equal and called as **semicircle.**

**Eg:-** In figure below P and Q are ends of diameter PQ. Therefore, PRQ and PSQ are equal and semicircle.

**(vi) Segment-**

The region between the chord and the arc is called as **segment **of the circle.

**Eg:-** In figure below, circle has two segments.

**Types of the segments –**

**(a) Major segment-**

The region between the chord and the large arc is called as **major segment** of the circle.

**Eg:-** In figure below PRQ is a major segment.

**(b) Minor segment-**

The region between the chord and the small arc is called as** minor segment** of the circle.

**Eg:-** In figure below PSQ is a minor segment.

**(vii)Sector-**

The region between an arc and two radii is known as **sector** of a circle.

**Eg:-** In figure below, circle has two sectors.

**Types of sector-**

**(a) Major sector-**

The region between major arc and two radii is known as **major sector** of a circle.

**Eg:- **In figure below, circle has major sector.

**(b)Minor sector-**

The region between minor arc and two radii is known as **minor sector** of a circle.

**Eg:-** In figure below, circle has minor sector.

**(2) **A circle divides a plane on which it lies into **three parts.**

**(i)**The region of plane inside of the circle is known as **interior of the circle**.

**Eg:-** In figure below, interior of the circle is presented.

**(ii)**The region of plane outside of the circle is known as **exterior of the circle**.

**Eg:-** In figure below, exterior of the circle is presented.

**(iii)**The boundary of the circle is known as circle.

**Eg:-** In figure below, circle is presented.

The circle and its interior make up the circular region.

**Take three non – collinear points A, B and C.**

**Draw perpendicular bisectors PQ and RS of AB and BC respectively.**

**Let these perpendiculars bisectors intersect at point O. ( Two lines intersect only at one point.)**

**Point O lies on the perpendicular bisector PQ of AB,**

** OA = OB …(1) **

**(Every point of perpendicular PQ is equidistance from end points of AB)**

**Similarly, point O lies on the perpendicular bisector RS of BC,**

** OB = OC ( every point of perpendicular PQ is equidistance from end points of AB)…(2)**

** OA = OB = OC (From equation 1 and 2)**

**Therefore, we can draw a circle with centre O which will pass through point A, B and C.**

**Hence, proved.**

**Helping Topics**