CAT Introduction to Circles - Practice Questions & MCQ

As we all know that the invention of the wheel revolutionized the entire human life. In fact without wheels no means of transportation is possible, starting from bullock cart to aeroplane.

In daily life, we may have come across many objects which are round in shape, such as wheels of a vehicle, bangles, dials of many clocks, coins of denominations 1, 2 and 5, etc. In this chapter, we will study about circles, other related terms, and some properties of a circle.

You can easily trace out a circle using a compass and a pencil. 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. The closed figure traced by the pencil on paper by you is a circle.

Important Definitions

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: The fixed point is called the centre of the circle. In the figure, O is the centre.

Radius: The segment joining the centre and a point on the circle is called its radius. The plural of radius is radii. In figure the length OP is the radius of the circle.

Interior and Exterior of Circle

A circle divides the plane on which it lies into three parts. They are:

1. inside the circle, which is also called the interior of the circle;
2. the circle and
3. outside the circle, which is also called the exterior of the circle

The circle and its interior make up the circular region.

Chord: A line segment joining two points on a circle is called a chord of the circle. In the figure, AB is a chord of the circle.

Diameter: The chord, which passes through the centre of the circle, is called a diameter of the circle. In the figure, AB is a chord of the circle. A diameter of a given circle is the largest chord of the circle.

The word diameter is used for a chord passing through the centre and also, for its length. If d is the diameter of the circle then d = 2r where r is the radius.

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

Let P and Q be two points on the circle. These points P and Q divide the circle into two parts. Each part is an arc. You find that there are two pieces, one longer and the other smaller. 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. Unless otherwise stated, arc PQ or stands for minor arc PQ.

If the length of an arc is less than the length of the arc of the semicircle then it is called a minor arc. Otherwise, it is a major arc.

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

Degree Measure of an Arc

The degree measure of a minor arc is the measure of the central angle subtended by the arc.

Let  be an arc of a circle with centre O. If  then degree measure of . And, we write,.

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.

Again, there are two types of segments, which are the major segment and the minor segment. The segment, containing the minor arc, is called a minor segment and the segment, containing the major arc, is called the major segment. 

Sector: The region between an arc and the two radii, joining the centre to the endpoints of the arc is called a sector. Like segments,  the minor arc corresponds to the minor sector and the major arc corresponds to 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.

Concentric Circles: Circles having the same centre but different radius are said to be concentric circles.

Congruent Circles: Two circles C(O, r) and C(O', s) are congruent if they have the same radii i.e. r = s.