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title: "I’m Up and Down, and Round and Round"
board: "CBSE"
curriculum: "CBSE"
class: "Class 9"
subject: "Mathematics"
book: "Ganita Manjari"
chapter: "I’m Up and Down, and Round and Round"
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---

# I’m Up and Down, and Round and Round
Humanity has always been fascinated by the shapes of the things around them. In some early cave paintings, the sun is depicted as a circle. Circles form when raindrops fall on water, and various natural objects like cross-sections of plant stems and sunflowers have circular shapes. This chapter explores the properties and definitions of circles.

---

## Knowledge Snapshot

| Field | Details |
| :--- | :--- |
| Class | Class 9 |
| Subject | Mathematics |
| Book | Ganita Manjari |
| Chapter | I’m Up and Down, and Round and Round |
| Pages | 92-117 |

---

## Chapter Summary

### Short Summary
This chapter introduces the concept of circles, defining them as the set of all points equidistant from a center point in a two-dimensional plane. It discusses various properties, symmetries, and characteristics of circles, accompanied by engaging activities and exercises.

### Detailed Summary
The chapter begins with a historical context, illustrating humanity's fascination with circular shapes in nature. It defines a circle as the locus of points at a constant distance (radius) from a central point. Various properties are discussed, including symmetry—both rotational and reflective—and the concept of diameters and chords. The chapter further investigates how many circles can be drawn through specific points, establishing the uniqueness of circles through three non-collinear points and explaining the role of circumcentres. It also details the relationships between chords, their angles, and the positions relative to the circle. Theorems regarding the angles subtended by arcs and properties of cyclic quadrilaterals are established, concluding with exercises to reinforce the learned concepts.

---

## Topic-Wise Explanation

### Definitions of Circle and Chord
A circle is defined as the set of all points on a plane that are equidistant from a given point, known as the center. A chord is defined as a line segment joining two points on a circle.

### Symmetries of a Circle
Circles exhibit complete rotational symmetry; when rotated by any angle, they appear unchanged. Additionally, all diameters are lines of reflection symmetry.

### How Many Circles?
There are infinitely many circles that can pass through two given points on a plane, with their centers lying on the perpendicular bisector of the segment that connects those points.

### Chords and the Angles They Subtend
Equal chords subtend equal angles at the center, and conversely, if two chords subtend equal angles, they are equal in length.

### Midpoints and Perpendicular Bisectors of Chords
The line segment from the center to the midpoint of a chord is perpendicular to the chord. This leads to further understandings of the properties regarding the distances of chords from the center of the circle.

### Distance of Chords from the Centre
Chords of equal length are equidistant from the center of a circle. Conversely, if two chords are equidistant from the center, they are equal in length.

### Angles Subtended by an Arc
The angle subtended by an arc at the center is twice the angle subtended by that arc at any point on the circle outside the arc.

### Concyclicity of Points
Four points that subtend equal angles at a line segment are concyclic, meaning they lie on the same circle. A unique circle can pass through any three non-collinear points.

---

## Core Ideas

| Idea | Explanation |
| :--- | :--- |
| Circle Definition | A circle is the locus of points equidistant from a center point. |
| Chord Definition | A chord connects two points on the circle. |
| Angle Subtended | The angle at the center is twice that at the circumference for the same arc. |

---

## Key Concepts

| Concept | Meaning |
| :--- | :--- |
| Circumcentre | Center of the circumcircle that passes through the vertices of a triangle. |
| Diameter | A chord that passes through the center of the circle. |
| Radius | The distance from the center to any point on the circle. |

---

## Important Points for Revision
* A circle is defined as the set of points equidistant from a central point.
* Every circle has a center and radius.
* Equal chords subtend equal angles at the center.
* The diameter is the longest chord in a circle.
* If two chords are equidistant from the center, they are of equal length.
* The angle subtended by a diameter at any point on the circle is 90°.
* Four points that subtend equal angles at a line segment are concyclic.
* The sum of opposite angles in a cyclic quadrilateral is 180°.

---

## Practice Questions

### Short Answer Questions
1. Define a circle.
2. What is a chord?
3. State the properties of equal chords.
4. How many circles can be drawn through two points?
5. What is the rule for angles subtended by arcs?

### Long Answer Questions
1. Prove that the perpendicular bisector of a chord passes through the center of the circle.
2. Explain the relationship between diameters and lines of symmetry in circles.
3. Illustrate how to find the circumcentre of a triangle formed by three non-collinear points.

---

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