📘 About the new NCERT Class 9 book Ganita Manjari
The new NCERT Maths book “Ganita Manjari” ✨ is designed to make mathematics more interactive, meaningful, and connected to real life 🧠🌍. Instead of only focusing on formulas, it:
- Uses stories and situations (like Reiaan’s room 🏠) to explain concepts
- Encourages thinking, reasoning, and exploration 🤔
- Includes “Think and Reflect” questions 💭 to build deeper understanding
- Connects maths with history and practical use 📜📏
👉 Overall, this new book focuses less on rote learning and more on understanding concepts clearly and applying them in real situations 🌐.
⭕ Chapter 5: I’m Up and Down, and Round and Round
CBSE Class 9 – Ganita Manjari
🌟 This chapter explores the beautiful world of circles. You will learn about the definition of a circle, its symmetry, chords, arcs, angles, distances from the centre, circumcircles, concyclic points, and cyclic quadrilaterals.
🎯 Learning Objectives
- ⭕ Understand the definition and basic elements of a circle.
- 📏 Learn about radius, diameter and chord.
- 🔄 Understand rotational and reflection symmetry.
- 📐 Study the relationship between chords and angles.
- 📍 Understand the distance of a chord from the centre.
- 🌙 Learn about arcs and angles subtended by them.
- 🔵 Understand circumcircles and circumcentres.
- 🔗 Learn when points are concyclic.
- ⬛ Understand important properties of cyclic quadrilaterals.
⭕ 1. Definition of a Circle
A circle is the set of all points on a plane that are at an equal distance from a fixed point.
The fixed point is called the centre and the equal distance is called the radius.
Radius = Distance from the centre to any point on the circle.
🔑 Important Terms
- Centre: The fixed point of a circle.
- Radius: Distance from the centre to the circle.
- Chord: A line segment joining two points on a circle.
- Diameter: A chord passing through the centre.
- Locus: The set of points satisfying a particular condition.
🔄 2. Symmetries of a Circle
A circle has complete rotational symmetry. It looks the same after rotation through any angle.
Every diameter of a circle is also a line of reflection symmetry.
🔵 3. Circles Through Two and Three Points
Given two points A and B, the centres of all circles passing through A and B lie on the perpendicular bisector of AB.
Three non-collinear points determine exactly one circle. This circle is called the circumcircle.
📐 Circumcentre
The centre of the circumcircle of a triangle is called its circumcentre.
- 🔺 In an acute-angled triangle, the circumcentre lies inside.
- 🔺 In an obtuse-angled triangle, the circumcentre lies outside.
- 🔺 In a right-angled triangle, the circumcentre is the midpoint of the hypotenuse.
📏 4. Chords and Angles They Subtend
Chords have important relationships with the angles they subtend at the centre of a circle.
📌 Theorem 1
Equal chords of a circle subtend equal angles at the centre.
📌 Theorem 2
Chords that subtend equal angles at the centre are equal.
These results are commonly proved using congruence of triangles.
📍 5. Midpoints and Perpendicular Bisectors of Chords
📌 Important Theorem
The line joining the centre of a circle to the midpoint of a chord is perpendicular to the chord.
📌 Converse
The perpendicular from the centre of a circle to a chord bisects the chord.
Therefore, if a perpendicular is drawn from the centre to a chord, it divides the chord into two equal parts.
📐 6. Distance of Chords from the Centre
The chapter establishes an important relationship between the length of a chord and its distance from the centre.
Another important result is:
📌 Shorter chord → Farther from the centre
The diameter is the longest chord of a circle because it passes through the centre.
🧮 Chord Length Formula
If the radius of a circle is r and the perpendicular distance of a chord from the centre is d, then:
🌙 7. Arcs and Angles Subtended by an Arc
An arc is a connected portion of the circumference of a circle between two points.
- 🌙 Minor Arc: The smaller part of the circle.
- 🌕 Major Arc: The larger part of the circle.
📌 Major Theorem
The angle subtended by an arc at the centre of a circle is twice the angle subtended by the same arc at any point on the circle outside the arc.
Central angle = 2 × Angle at the circumference
⭐ 8. Angle in a Semicircle
📌 Key Result
The angle subtended by a diameter at any point on the circle is 90°.
This is one of the most useful results for solving problems involving circles and triangles.
🔗 9. Concyclicity of Points
Points that lie on the same circle are called concyclic points.
📌 Concyclicity Theorem
If a line segment AB subtends equal angles at two points C and D lying on the same side of AB, then A, B, C and D are concyclic.
⬛ 10. Cyclic Quadrilateral
A quadrilateral whose four vertices lie on the same circle is called a cyclic quadrilateral.
📌 Important Theorem
The sum of two opposite angles of a cyclic quadrilateral is 180°.
∠B + ∠D = 180°
🔄 Converse
If the opposite angles of a quadrilateral add up to 180°, then the four vertices are concyclic.
📚 Important Theorems at a Glance
| 📌 Theorem | 📝 Result |
|---|---|
| 1 | Three non-collinear points determine a unique circle. |
| 2 | Equal chords subtend equal angles at the centre. |
| 3 | Equal central angles correspond to equal chords. |
| 4 | Line joining centre to midpoint of a chord is perpendicular to the chord. |
| 5 | Perpendicular from centre to a chord bisects the chord. |
| 6 | Equal chords are equidistant from the centre. |
| 7 | Chords equidistant from the centre are equal. |
| 8 | Longer chord is closer to the centre. |
| 9 | Central angle is twice the angle subtended at the circumference. |
| 10 | Equal angles subtended by the same segment can establish concyclicity. |
| 11 | Opposite angles of a cyclic quadrilateral are supplementary. |
| 12 | If opposite angles of a quadrilateral sum to 180°, it is cyclic. |
🧠 Quick Revision
⭕ Circle: Set of points at equal distance from a fixed point.
📏 Diameter: Longest chord.
📐 Central angle: Twice the angle at the circumference for the same arc.
📍 Equal chords: Equidistant from the centre.
📏 Longer chord: Closer to the centre.
⭐ Angle in a semicircle: 90°.
🔗 Concyclic points: Points lying on the same circle.
⬛ Cyclic quadrilateral: Opposite angles add up to 180°.
❓ Frequently Asked Questions – FAQs
1️⃣ 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 called the centre.
2️⃣ What is the longest chord of a circle?
The diameter is the longest chord because it passes through the centre of the circle.
3️⃣ What is the relationship between equal chords and central angles?
Equal chords of a circle subtend equal angles at the centre. Conversely, equal central angles correspond to equal chords.
4️⃣ What happens when a perpendicular is drawn from the centre to a chord?
The perpendicular from the centre to a chord bisects the chord.
5️⃣ Are equal chords equally distant from the centre?
Yes. Equal chords of a circle are at equal perpendicular distances from the centre.
6️⃣ Which chord is closer to the centre: a longer chord or a shorter chord?
The longer chord is closer to the centre, while the shorter chord is farther away.
7️⃣ What is the angle in a semicircle?
The angle subtended by a diameter at any point on the circle is 90°.
8️⃣ What are concyclic points?
Points that lie on the same circle are called concyclic points.
9️⃣ What is a cyclic quadrilateral?
A quadrilateral whose four vertices lie on the same circle is called a cyclic quadrilateral.
🔟 What is the property of opposite angles of a cyclic quadrilateral?
The opposite angles of a cyclic quadrilateral are supplementary, meaning their sum is 180°.