---
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id: "69f86a06293c3123114dbb21"
title: "Circles"
board: "CBSE"
curriculum: "CBSE"
class: "Class 10"
subject: "Mathematics"
book: "Mathematics"
chapter: "Circles"
chapter_slug: "circles"
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---

# Circles
In this chapter, we explore the concept of circles, their properties, and the relationship between lines and circles, specifically tangents and secants. A circle consists of all points in a plane that are at a constant distance from a fixed point known as the center.

---

## Knowledge Snapshot

| Field | Details |
| :--- | :--- |
| Class | Class 10 |
| Subject | Mathematics |
| Book | Mathematics |
| Chapter | Circles |
| Pages | 144-153 |

---

## Chapter Summary

### Short Summary
This chapter covers the definition of a circle, the properties of tangents, and how they relate to the radius and the characteristics of secants and tangents related to circle geometry.

### Detailed Summary
The chapter provides a fundamental understanding of circles, addressing the definitions of key terms such as tangents and secants. It includes methods to determine the existence of tangents to a circle and illustrates the angle relationships and properties of tangents and their interactions with radii and other lines.

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## Topic-Wise Explanation

### Introduction to Circles
A circle is defined as a collection of points equidistant from a center point. The chapter introduces the possible interactions between a line and a circle: non-intersecting line, secant, and tangent.

### Tangent to a Circle
A tangent intersects the circle at exactly one point. Fundamental activities demonstrate that as a line approaches a tangent position, the intersection points converge to a single point.

### Existence of Tangents
The existence of tangents is explored through geometric constructions. Key observations are made regarding the number of tangents that can be drawn from various points relative to the circle.

### Theorem 10.1: Tangent and Radius Relationship
The chapter proves that a tangent to a circle is perpendicular to the radius at the point of contact.

### Number of Tangents from a Point on a Circle
The chapter outlines three scenarios regarding tangent lines from points inside, on, and outside a circle, emphasizing the conditions for drawing tangents.

### Theorem 10.2: Equal Lengths of Tangents
This theorem states that the lengths of tangents from an external point to a circle are equal, demonstrated through triangle congruence.

---

## Core Ideas

| Idea | Explanation |
| :--- | :--- |
| Definition of a Circle | Collection of all points equidistant from a center point. |
| Tangent Line | A line that touches the circle at exactly one point. |
| Secant Line | A line that intersects the circle at two points. |
| Tangent Properties | A tangent is perpendicular to the radius at the point of contact. |

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## Key Concepts

| Concept | Meaning |
| :--- | :--- |
| Tangent | A line that touches a circle at one point. |
| Secant | A line that intersects a circle at two points. |
| Radius | Distance from the center of the circle to any point on the circle. |

---

## Important Points for Revision

* A circle's definition includes all points at a constant distance from the center.
* A tangent intersects the circle at exactly one point.
* A secant intersects the circle at two points.
* There can be exactly one tangent at any point on a circle.
* The tangent to a circle is perpendicular to the radius at the point of contact.
* A point inside a circle has no tangents that can be drawn.
* A point on the circle allows for exactly one tangent line.
* A point outside the circle allows for exactly two tangent lines.
* The lengths of tangents from an external point to a circle are equal.
* The parallel tangents are established through secant lines.

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## Practice Questions

### Short Answer Questions
1. Define a tangent to a circle.
2. How many tangents can be drawn from a point inside a circle?
3. Explain the difference between a secant and a tangent.
4. What is the relationship between a tangent and the radius at the point of contact?
5. How many tangents can be drawn from an external point to a circle?

### Long Answer Questions
1. Prove that the tangent at the end of a diameter is parallel to the opposite side of the circle.
2. Describe the construction of a circle with two tangents drawn from an external point and derive relevant relations.
3. Explain and prove Theorem 10.1 concerning tangents and radii.
4. Illustrate the proof of Theorem 10.2 using a geometric construction.

---

## Related Concepts

| Concept | Details |
| :--- | :--- |
| Radius | Important measure in circle geometry. |
| Chord | A line segment with both endpoints on the circle. |
| Concentric Circles | Circles sharing the same center but different radii. |

---

## Source Attribution

| Field | Value |
| :--- | :--- |
| Source | Edzy |
| Reference Type | examSubjectBookChapter |
| Reference ID | 69f86a06293c3123114dbb21 |
| Canonical URL | https://www.edzy.ai/cbse-class-10-mathematics-circles |
| Markdown URL | https://www.edzy.ai/okf/chapter/cbse-class-10-mathematics-circles.md |