CBSE Class 9 Maths Revision Notes Chapter 5 I’m Up and Down, and Round and Round

A circle is the locus of all points in a plane that remain at the same distance from a fixed point called the centre. Its chords, arcs and angles follow important geometric relationships used in proofs and numerical problems.

The chapter I’m Up and Down, and Round and Round develops the main geometric properties of circles. It explains how circles are formed, how chords behave and how angles made by arcs are related.

These CBSE Class 9 Maths Revision Notes Chapter 5 cover circumcircles, chord theorems, concyclic points and cyclic quadrilaterals. The notes follow the current 2026–27 textbook.

Key Takeaways

• Circle: Set of all points at a fixed distance from the centre.
• Diameter: Longest chord of a circle and equal to twice the radius.
• Arc theorem: The angle at the centre is twice the angle at the circle on the same arc.
• Cyclic quadrilateral: Its opposite angles add up to 180°.

Access Class 9 Maths Chapter 5 Notes in 30 Minutes

Use this order for quick revision:

First 10 minutes: Circle definitions, chords and circumcircle
Next 10 minutes: Chord theorems and chord-length problems
Final 10 minutes: Arc angles, concyclic points and cyclic quadrilaterals

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Circle Basics in Class 9 Maths Chapter 5 Notes

A circle is one of the most symmetric geometric figures. Its main parts are defined using the centre and points on the circle.

What Is a Circle?

A circle is the set of all points in a plane that are at the same distance from a fixed point.

The fixed point is called the centre.

The fixed distance is called the radius.

A circle may also be described as the locus of points equidistant from a given point.

Important Parts of a Circle

Chord and Diameter

Every diameter is a chord.

However, every chord is not a diameter.

The diameter is the longest possible chord of a circle.

If the radius is r, then:

Diameter = 2r

Symmetry of a Circle

A circle has complete rotational symmetry.

It looks unchanged after rotation through any angle around its centre.

Every diameter is also a line of reflection symmetry.

Therefore, a circle has infinitely many lines of symmetry.

Circles Through Given Points in CBSE Class 9 Maths Chapter 5 Notes

The number of circles that can pass through given points depends on the position of those points.

How Many Circles Pass Through One Point?

Infinitely many circles can pass through one point.

The centre may lie at different positions, provided the radius reaches the given point.

How Many Circles Pass Through Two Points?

Infinitely many circles can pass through two distinct points A and B.

The centre of each such circle lies on the perpendicular bisector of AB.

This is because every point on the perpendicular bisector is equidistant from A and B.

Smallest Circle Through Two Points

The smallest circle through A and B has AB as its diameter.

Its centre is the midpoint of AB.

Its radius is:

AB/2

There is no largest circle through the two points because the centre can move farther along the perpendicular bisector.

Circle Through Three Points

Three collinear points cannot lie on one circle.

A straight line cannot intersect a circle at three distinct points.

However, exactly one circle passes through three non-collinear points.

Theorem: Unique Circle Through Three Non-Collinear Points

A unique circle can be drawn through any three non-collinear points.

Reason

Let the points be A, B and C.

• The centre must lie on the perpendicular bisector of AB.
• It must also lie on the perpendicular bisector of AC.
• These two perpendicular bisectors meet at one point.

That point is equidistant from A, B and C.

Therefore, it becomes the unique centre of the required circle.

Circumcentre and Circumcircle Class 9 Notes

A circle passing through all three vertices of a triangle is called its circumcircle.

The centre of this circle is called the circumcentre.

How to Construct the Circumcentre

1. Draw the perpendicular bisector of one side.
2. Draw the perpendicular bisector of another side.
3. Mark their point of intersection.
4. This point is the circumcentre.
5. Use it as the centre and the distance to any vertex as radius.

Position of the Circumcentre

For a right triangle, the hypotenuse acts as the diameter of its circumcircle.

Chord Theorems in Class 9 Maths Revision Notes Chapter 5

Chords follow several important properties related to the centre and central angles.

Equal Chords and Equal Angles at the Centre

Theorem 1

Equal chords of the same circle subtend equal angles at the centre.

If:

AB = CD

then:

∠AOB = ∠COD

where O is the centre.

Proof Idea

Join the endpoints of both chords to the centre.

In triangles AOB and COD:

• OA = OC, as radii
• OB = OD, as radii
• AB = CD, given

Therefore, the triangles are congruent by SSS.

Hence, the central angles are equal.

Equal Central Angles and Equal Chords

Theorem 2

Chords that subtend equal angles at the centre are equal.

If:

∠AOB = ∠COD

then:

AB = CD

The proof uses SAS congruence.

Centre and Midpoint of a Chord

Theorem 3

The line joining the centre of a circle to the midpoint of a chord is perpendicular to the chord.

If M is the midpoint of chord AB, then:

OM ⟂ AB

Proof Idea

In triangles OMA and OMB:

• OA = OB
• AM = MB
• OM is common

The triangles are congruent by SSS.

The two adjacent angles at M are equal and form a straight angle.

Therefore, each is 90°.

Perpendicular from Centre to Chord

Theorem 4

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

If:

OM ⟂ AB

then:

AM = MB

This is the converse of the previous theorem.

Equal Chords and Distance from Centre

The distance of a chord from the centre means the perpendicular distance.

Theorem 5

Equal chords of a circle are equidistant from the centre.

If:

AB = CD

then:

Distance of AB from O = Distance of CD from O

Theorem 6

Chords equidistant from the centre are equal.

These two theorems are converses of each other.

Unequal Chords and Distance from Centre

Theorem 7

The longer chord is closer to the centre.

If:

AB > CD

then:

Distance of AB from O < Distance of CD from O

The diameter is the nearest possible chord to the centre because its distance is zero.

It is also the longest chord.

Formula for Chord Length

Suppose:

• Radius = r
• Perpendicular distance from centre to chord = d

The perpendicular bisects the chord.

Using the Pythagoras theorem:

Half chord = √(r² − d²)

Therefore:

Chord length = 2√(r² − d²)

Example 1

Radius = 7 cm
Distance from centre = 6 cm

Chord length = 2√(7² − 6²)

= 2√(49 − 36)

= 2√13 cm

Arcs and Angles in Circle Notes

An arc is a connected portion of the circle between two points.

Minor Arc

A minor arc is the smaller arc between two points.

It subtends an angle less than 180° at the centre.

Major Arc

A major arc is the larger arc between two points.

It subtends an angle greater than 180° at the centre.

Angle Subtended by an Arc at the Centre

If arc AB has centre O, then the angle formed by radii OA and OB is the angle subtended by the arc at the centre.

This angle is:

∠AOB

Angle Subtended by an Arc at a Point on the Circle

Let P be a point on the circle outside arc AB.

Then the angle subtended by arc AB at P is:

∠APB

Theorem 8

The angle subtended by an arc at the centre is twice the angle subtended by the same arc at any point on the remaining circle.

Therefore:

∠AOB = 2∠APB

or:

∠APB = 1/2 ∠AOB

Example 2

If the central angle is 70°, then the angle at the circle is:

70°/2 = 35°

Angles in the Same Segment

Angles subtended by the same chord or arc at points on the same segment are equal.

If points P and Q lie on the same side of chord AB, then:

∠APB = ∠AQB

Angle in a Semicircle Class 9

A diameter subtends a straight angle of 180° at the centre.

Therefore, the angle subtended by a diameter at any point on the circle is:

180°/2 = 90°

So, the angle in a semicircle is always a right angle.

If AB is a diameter and P lies on the circle, then:

∠APB = 90°

Concyclic Points in Class 9 Maths Chapter 5 Notes

Points that lie on the same circle are called concyclic points.

Three non-collinear points are always concyclic because a unique circle passes through them.

For four points, an additional condition is required.

Concyclicity Criterion

If a line segment AB subtends equal angles at two points C and D on the same side of AB, then:

∠ACB = ∠ADB

Therefore, points A, B, C and D are concyclic.

Cyclic Quadrilateral Class 9 Notes

A quadrilateral whose four vertices lie on a circle is called a cyclic quadrilateral.

Theorem 9

The sum of opposite angles of a cyclic quadrilateral is 180°.

For cyclic quadrilateral ABCD:

∠A + ∠C = 180°

and:

∠B + ∠D = 180°

Example 3

If:

∠A = 75°

then:

∠C = 180° − 75°

= 105°

Converse Theorem

If the sum of a pair of opposite angles of a quadrilateral is 180°, then the quadrilateral is cyclic.

If:

∠A + ∠C = 180°

then points A, B, C and D lie on one circle.

Exterior Angle Property

The exterior angle of a cyclic quadrilateral equals the interior opposite angle.

For example, if side CD is extended to E, then:

∠CDE = ∠ABC

Important Circle Theorems Class 9

Solved Examples in Class 9 Maths Chapter 5

Example 4: Find Chord Length

Radius = 13 cm
Distance from centre = 5 cm

Chord length = 2√(13² − 5²)

= 2√(169 − 25)

= 2√144

= 24 cm

Example 5: Find Distance from Centre

Diameter = 26 cm

Radius = 13 cm

Chord length = 24 cm

Half chord = 12 cm

Let distance from centre be d.

Using the Pythagoras theorem:

13² = 12² + d²

169 = 144 + d²

d² = 25

d = 5 cm

Example 6: Cyclic Quadrilateral

In cyclic quadrilateral PQRS:

∠P = 2x + 10°

∠R = 3x − 20°

Opposite angles are supplementary:

2x + 10 + 3x − 20 = 180

5x − 10 = 180

5x = 190

x = 38

Therefore:

∠P = 86°

∠R = 94°

Common Mistakes in Circle Problems

Mistake 1: Treating Every Chord as a Diameter

A diameter must pass through the centre.

A chord may or may not pass through the centre.

Mistake 2: Forgetting the Half-Chord

The perpendicular from the centre bisects the chord.

Use half the chord while applying the Pythagoras theorem.

Mistake 3: Mixing Central and Inscribed Angles

The central angle is twice the angle at the circle on the same arc.

Mistake 4: Using Cyclic Properties Without Proving Concyclicity

Opposite angles are supplementary only when the quadrilateral is cyclic.

Mistake 5: Assuming Three Collinear Points Form a Circle

Three collinear points cannot lie on one circle.

Quick Revision of I’m Up and Down, and Round and Round

• A circle is a locus of points equidistant from a centre.
• The diameter is the longest chord.
• Every diameter is a line of symmetry.
• Infinitely many circles pass through two points.
• One unique circle passes through three non-collinear points.
• The circumcentre is found using perpendicular bisectors.
• Equal chords subtend equal central angles.
• A perpendicular from the centre bisects a chord.
• Equal chords are equidistant from the centre.
• Longer chords lie closer to the centre.
• Central angle is twice the angle at the circle.
• The angle in a semicircle is 90°.
• Concyclic points lie on one circle.
• Opposite angles of a cyclic quadrilateral add to 180°.

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