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• Introduction

• Circles and Its Related Terms: A Review

• Circles and Its Related Terms: A Review

You may have come across many objects in daily life, which are round in shape, such as wheels of a vehicle, bangles, dials of many clocks, coins of denominations 50 p, Re 1 and Rs 5, key rings, buttons of shirts, etc. (see Fig.10.1).

In a clock, you might have observed that the second’s hand goes round the dial of the clock rapidly and its tip moves in a round path. '

This path traced by the tip of the second’s hand is called a circle. In this chapter, you will study about circles, other related terms and some properties of a circle.

In a clock, you might have observed that the second’s hand goes round the dial of the clock rapidly and its tip moves in a round path. '

This path traced by the tip of the second’s hand is called a circle. In this chapter, you will study about circles, other related terms and some properties of a circle.

Take a compass and fix a pencil in it. Put its pointed leg on a point on a sheet of a paper. Open the other leg to some distance.

Keeping the pointed leg on the same point, rotate the other leg through one revolution. What is the closed figure traced by the pencil on paper? As you know, it is a circle (see Fig.10.2).

How did you get a circle? You kept one point fixed (A in Fig.10.2) and drew all the points that were at a fixed distance from A. This gives us the following definition:

`bb"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. In Fig.10.3, O is the centre and the length OP is the radius of the circle.

`"Remark :"`The line segment joining the centre and any point on the circle is also called a radius of the circle. That is, ‘radius’ is used in two senses-in the sense of a line segment and also in the sense of its length.

You are already familiar with some of the following concepts. We are just recalling them.

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.10.4).

The circle and its interior make up the circular region.

If you take two points P and Q on a circle, then the line segment PQ is called a chord of the circle (see Fig. 10.5).

The chord, which passes through the centre of the circle, is called a diameter of the circle. As in the case of radius, the word ‘diameter’ is also used in two senses, that is, as a line segment and also as its length.

Do you find any other chord of the circle longer than a diameter? No, you see that a diameter is the longest chord and all diameters have the same length, which is equal to two times the radius. In Fig.10.5, AOB is a diameter of the circle.

How many diameters does a circle have? Draw a circle and see how many diameters you can find.

A piece of a circle between two points is called an arc. Look at the pieces of the circle between two points P and Q in Fig.10.6.

You find that there are two pieces, one longer and the other smaller (see Fig.10.7). The longer one is called the major arc PQ and the shorter one is called the minor are PQ. The minor arc PQ is also denoted by PQ and the major arc PQ by PRQ , where R is some point on the arc between P and Q.

Unless otherwise stated, arc PQ or PQ stands for minor arc PQ. When P and Q are ends of a diameter, then both arcs are equal and each is called a semicircle.

The length of the complete circle is called its circumference. 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.

You will find that there are two types of segments also, which are the major segment and the minor segment (see Fig. 10.8). T

he region between an arc and the two radii, joining the centre to the end points of the arc is called a sector. Like segments, you find that the minor arc corresponds to the minor sector and the major arc corresponds to the major sector.

In Fig. 10.9, the region OPQ is the minor sector and 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.

Keeping the pointed leg on the same point, rotate the other leg through one revolution. What is the closed figure traced by the pencil on paper? As you know, it is a circle (see Fig.10.2).

How did you get a circle? You kept one point fixed (A in Fig.10.2) and drew all the points that were at a fixed distance from A. This gives us the following definition:

`bb"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. In Fig.10.3, O is the centre and the length OP is the radius of the circle.

`"Remark :"`The line segment joining the centre and any point on the circle is also called a radius of the circle. That is, ‘radius’ is used in two senses-in the sense of a line segment and also in the sense of its length.

You are already familiar with some of the following concepts. We are just recalling them.

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.10.4).

The circle and its interior make up the circular region.

If you take two points P and Q on a circle, then the line segment PQ is called a chord of the circle (see Fig. 10.5).

The chord, which passes through the centre of the circle, is called a diameter of the circle. As in the case of radius, the word ‘diameter’ is also used in two senses, that is, as a line segment and also as its length.

Do you find any other chord of the circle longer than a diameter? No, you see that a diameter is the longest chord and all diameters have the same length, which is equal to two times the radius. In Fig.10.5, AOB is a diameter of the circle.

How many diameters does a circle have? Draw a circle and see how many diameters you can find.

A piece of a circle between two points is called an arc. Look at the pieces of the circle between two points P and Q in Fig.10.6.

You find that there are two pieces, one longer and the other smaller (see Fig.10.7). The longer one is called the major arc PQ and the shorter one is called the minor are PQ. The minor arc PQ is also denoted by PQ and the major arc PQ by PRQ , where R is some point on the arc between P and Q.

Unless otherwise stated, arc PQ or PQ stands for minor arc PQ. When P and Q are ends of a diameter, then both arcs are equal and each is called a semicircle.

The length of the complete circle is called its circumference. 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.

You will find that there are two types of segments also, which are the major segment and the minor segment (see Fig. 10.8). T

he region between an arc and the two radii, joining the centre to the end points of the arc is called a sector. Like segments, you find that the minor arc corresponds to the minor sector and the major arc corresponds to the major sector.

In Fig. 10.9, the region OPQ is the minor sector and 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.