ICSE Class 8 Maths Chapter 30 Circles

Chapter 30

Circles

30.1 Introduction

Circle

A circle is a plane figure obtained when a point moves in such a way that it is always at same distance from a fixed point.

The adjoining figure shows a fixed point O and a moving point A. The point A moves in such a way that it is always at the same (constant) distance from the fixed point O.

Clearly, the figure obtained (shown by the shaded region) is a circle.

30.2 Some Important Terms

1. Centre:

The fixed point is called the centre. In the given figure the fixed point O is the centre.

2. Radius:

The fixed (constant) distance of the moving point from the centre is called radius. In the given figure, the fixed distance is OA, so radius = OA.

3. Circumference:

The length of the boundary of a circle is called its circumference.

4. Chord:

The line segment joining any two points on the circumference of the circle is called a chord. In the given figure, PQ and CD are the chords of the given circle.

5. Diameter:

The chord of the circle, which passes through the centre, is called its diameter. In the given figure, the chord CD passes through the centre of the circle, so CD is a diameter of the circle.

Diameter is the largest chord of the circle.

Diameter = 2 × Radius and Radius = Diameter/2

30.3 Interior And Exterior Of A Circle

1. A point is said to be on the circumference of the circle if its distance from the centre is equal to the radius of the circle. In the given figure, the point A lies on the circumference of the circle; therefore, OA = r (the radius of the circle).

2. Interior of the circle:

The region inside the circle is called interior of the circle. The unshaded portion of the given circle (as shown above) represents the interior of the given circle.

The distance of every point in the interior of a circle from its centre is always less than its radius.

In the figure, given above, the point B lies in the interior of the circle, therefore OB is less than the radius i.e. OB < r.

3. Exterior of the circle:

The region outside the circle is called exterior of the circle. The shaded portion of the given figure (as shown on previous page) represents the exterior of the given circle.

The distance of every point in the exterior of a circle from its centre is always greater than its radius.

In the figure given on previous page, the point C lies in the exterior of the circle, therefore OC is greater than the radius i.e. OC > r.

Example 1:

P is a fixed point in a plane and a point Q moves in the same plane such the PQ is always 8 cm. State:

(i) the name of the figure formed.

(ii) the value of the radius of the figure formed.

(iii) the value of the diameter of the figure formed.

Can a chord of length 20 cm be drawn in this circle? Give reason.

Solution:

(i) The figure formed is a circle. (Ans.)

(ii) The value of the radius = The distance between the fixed point P and the moving point Q = PQ = 8 cm (Ans.)

(iii) The diameter of the figure (circle) formed = 2 × Radius = 2 × 8 cm = 16 cm (Ans.)

Since, the diameter is the largest chord of the circle and the diameter of the circle formed is 16 cm, therefore a chord of length 20 cm cannot be drawn in this circle. (Ans.)

Example 2:

The diameter of a circle is 24 cm. Find its radius. If O is the centre of the circle, state, giving reasons, the position of points P, Q and R, if:

(i) OP = 18 cm (ii) OQ = 12 cm (iii) OR = 9 cm.

Solution:

Since, diameter = 2 × radius

Radius = Diameter/2 = 24 cm/2 = 12 cm (Ans.)

(i) Since, OP = 18 cm and radius = 12 cm

=> The distance of the point P from the centre O is greater than the radius.

=> The point P lies in the exterior of the given circle. (Ans.)

(ii) Since, OQ = 12 cm

=> The distance of the point Q from the centre O is equal to the radius.

=> The point Q lies on the circumference of the given circle. (Ans.)

(iii) Since, OR = 9 cm

=> The distance of the point R from the centre O is less than the radius.

=> The point R lies in the interior of the given circle. (Ans.)

Teacher's Note

Understanding circles helps us recognize the shape of wheels, plates, and coins in everyday life. The radius concept is essential for tasks like adjusting compasses or setting the reach of sprinklers.

30.4 Arcs And Types Of Arcs

1. Arc:

A part of the circumference of a circle is called its arc. In the adjoining figure, APB is an arc of the given circle.

2. Semi-circle:

A diameter of a circle divides the circumference of the circle into two equal arcs.

In other words, a diameter of a circle bisects the circumference of the circle. Each of the two arcs so obtained, is called a semi-circle. In the given figure, arc APB is a semi-circle and arc AQB is also a semi-circle.

3. Major and minor arcs:

Let the circumference of circle be divided into two unequal parts.

The part of the circumference, which is greater than the semi-circle is called major arc and the part of the circumference, which is smaller than the semi-circle, is called minor arc.

In the given figure, the circumference of the circle is divided into two unequal parts APB and AQB. The arc AQB is greater than the semi-circle and the arc APB is smaller than the semi-circle.

Arc AQB = Major arc and arc APB = Minor arc

30.5 Sectors

The part of a circle, bounded by two radii and an arc, is called the sector of the circle.

The shaded portion of the given figure shows a sector as it is bounded by two radii OA and OB; and a minor arc APB.

Similarly, the unshaded portion of the given figure also shows a sector as it is also bounded by two radii OA and OB; a major arc AQB.

The part of the circle bounded by two radii and a minor arc is called the minor sector and the part of the circle bounded by two radii and a major arc is called the major sector.

In the circle, given above:

Minor arc = the shaded portion of the circle, and major arc = the unshaded portion of the circle.

30.6 Segments

A chord of a circle divides the given circle into two parts and each part so obtained is called a segment.

The segment bounded by the chord and the minor arc is called the minor segment, whereas the segment bounded by the chord and the major arc is called the major segment.

In the given figure, chord AB divides the circle into two unequal parts. The shaded portion of the circle is bounded by the chord AB and the minor arc APB, therefore the shaded portion represents the minor segment.

The unshaded portion of the circle is bounded by the chord AB and the major arc AQB, therefore the unshaded portion represents the major arc.

30.7 Central Angle

In a circle, the angle subtended by an arc (or a chord) at the centre of the circle, is called central angle.

In the given figure, arc ATB subtends angle AOB at the centre, therefore angle AOB is the central angle. Similarly, chord PQ subtends angle POQ at the centre, therefore angle POQ is the central angle.

30.8 Tangents

A line that meets (touches) the circle only at one point, is called tangent of the circle.

In the given figure, line segment AB touches the circle only at point P, so AB is tangent to the circle at point P.

The point P at which the tangent touches the circle is called the point of contact.

At the point of contact, the angle between the tangent and the radius is always 90 degrees, i.e. the tangent and the radius are perpendicular to each other.

Example 3:

O is the centre of a circle with radius 8 cm. P is a point outside the circle and PA is tangent of the circle. Find:

(i) the length of tangent PA, if OP = 10 cm

(ii) the distance between O and P, if the length of the tangent PA is 15 cm.

Solution:

According to the given information, the figure will be as shown alongside. Since, at the point of contact the angle between the tangent and the radius is 90 degrees, therefore angle OAP = 90 degrees.

(i) Given: Radius OA = 8 cm and OP = 10 cm

:. PA² + OA² = OP² [Using Pythagoras Theorem]

=> PA² + 8² = 10²

=> PA² = 100 - 64 = 36 = 6² => PA = 6 cm (Ans.)

(ii) Given: Radius OA = 8 cm and PA = 15 cm

:. OP² - OA² = PA² [Using Pythagoras Theorem]

=> OP² - 8² = 15²

=> OP² = 64 + 225 = 289 = 17² => OP = 17 cm (Ans.)

Teacher's Note

Tangent lines appear in real-world scenarios such as the path of a ball rolling off a curved surface or the trajectory of a spacecraft leaving orbital paths. Understanding tangents helps us analyze motion and forces in circular motion problems.

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