ICSE Class 10 Maths Chapter 17 Circles

Chapter 17 - Circles

Introduction

A circle is defined as the figure (closed curve) obtained by joining all those points in a plane which are at a constant distance from a fixed point in the same plane.

In fact, a circle is the locus of a point which moves in a plane in such a way that its distance from a fixed point in the same plane always remains constant.

The fixed point is called the centre of the circle and the constant distance is called the radius of the circle.

The perimeter of the circle is called its circumference.

Concentric Circles

Two or more circles are said to be concentric if they have same centre and different radii.

In the adjoining figure, O is the centre of each circle drawn; so the circles are called concentric circles.

Equal Circles

Circles are said to be equal or congruent if they have equal radii.

Circumscribed Circle

A circle that passes through all the vertices of a polygon is called the circumscribed circle. The centre of circumscribed circle is called circumcentre and the polygon is called inscribed polygon.

Inscribed Circle

A circle that touches all the sides of a polygon is called the inscribed circle (or, in-circle) of the polygon.

The centre of inscribed circle is called incentre and the polygon is called circumscribed polygon.

Chord

The line segment, joining any two points on the circumference of the circle, is called a chord.

A chord, which passes through the centre of the circle is called diameter, and is the largest chord of the circle.

Perpendicular From Centre To Chord

A straight line drawn from the centre of a circle to bisect a chord, which is not a diameter, is at right angles to the chord.

In the given figure, line OM, drawn from the centre O to bisect the chord AB, is perpendicular to AB.

i.e. AM = BM => OM ⊥ AB => ∠OMA = ∠OMB = 90°

Perpendicular Bisects The Chord

The perpendicular to a chord, from the centre of the circle, bisects the chord.

In the given figure, O is the centre of the circle and OP is perpendicular to the chord AB.

=> AP = BP

Remember

Greater is the size of a chord, smaller is its distance from the centre and vice-versa.

The adjoining figure shows a circle with centre O. OP is perpendicular to chord AB and OQ is perpendicular to chord CD. Since, chord AB is greater than chord CD => AB is at a smaller distance from the centre as compared to CD i.e. OP < OQ.

Conversely, as OP < OQ => AB > CD.

Circle Through Three Points

There is one circle, and only one, which passes through three given points not in a straight line.

Equal Chords Equidistant From Centre

Equal chords of a circle are equidistant from the centre.

Equidistant Chords Are Equal

Chords of a circle, equidistant from the centre of the circle, are equal.

In the given figure,

(i) if AB = CD => OM = ON

and, (ii) if OM = ON => AB = CD

Teacher's Note

Understanding circle properties helps in engineering and architecture, such as designing wheels, pipes, and circular structures in buildings.

Section 17.2 - Arc And Its Types

Arc Definition

An arc is a part of the circumference of a circle.

A chord divides the circumference of a circle into two parts and each part is called an arc.

In the figure, given alongside, chord AB divides the circumference into two unequal arcs APB and AQB.

The arc APB, which is less than the semi-circle, is called minor arc and the arc AQB, which is greater than the semi-circle, is called major arc.

Teacher's Note

Arcs are fundamental in measuring angles in circles and in calculating arc lengths, which are important in applications like road curves and circular tracks.

Section 17.3 - Segment And Relation Between Arcs And Segments

Segment Definition

A segment is the part of a circle bounded by an arc and a chord.

Theorem 5 - Angle At Centre Is Twice The Angle At Circumference

The angle which, an arc of a circle subtends at the centre is double that which it subtends at any point on the remaining part of the circumference.

Given: A circle with centre O. Arc APB subtends angle AOB at the centre and angle ACB at point C on the remaining circumference.

To Prove: ∠AOB = 2∠ACB.

Construction: Join CO and produce it to a point D.

Proof

Hence Proved.

Angles In Same Segment

Similarly, consider the following figures:

(i) ∠AOB = 2∠APB

(ii) Reflex ∠AOB = 2∠APB

Theorem 6 - Angles In Same Segment Are Equal

Angles in the same segment of a circle are equal.

Given: A circle with centre O. Angle ACB and angle ADB are in the same segment.

To Prove: ∠ACB = ∠ADB.

Construction: Join OA and OB.

Proof

Hence Proved.

Angles In Same Segment - Additional Cases

Similarly, in the adjoining figure:

(i) ∠DAB = ∠DCB [Angles in the same segment]

(ii) ∠ADC = ∠ABC [Angles in the same segment]

Theorem 7 - Angle In Semi-Circle Is A Right Angle

The angle in a semi-circle is a right angle.

Given: A circle with centre O. AB is a diameter and ACB is the angle of semi-circle.

To Prove: ∠ACB = 90°.

Teacher's Note

The angle in a semi-circle theorem is crucial for understanding right triangles inscribed in circles, which appears in navigation and architectural design.

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