### Circle:

The collection of all points in a plane which are at a fixed distance from a fixed point in the plane is called a circle.

### Chord -

The **line segment** within the circle joining any 2 points on the circle is called the chord.

### Diameter

– A **Chord** passing through the centre of the circle is called the **diameter.** – The **Diameter is 2 times the radius** and it is the **longest chord**.

### Arc

– The** portion** of a circle(curve)** between 2 points **is called an **arc**. – Among the two pieces made by an arc, the** longer** one is called a **major arc** and the **shorter** one is called a** minor arc.**

### Circumference

The **perimeter **of a circle is the **distance** covered by going around its** boundary once**. The perimeter of a circle has a special name: **Circumference**, which is π times the diameter which is given by the formula 2πr

**Semi-circle:**

A diameter of a circle divides it into two equal parts which an arc. Each of these two arcs is called a semi-circle.

**Congruent Circles (Arc):**

Two circles are said to be congruent if and only if either of them can be superposed on the other so as to cover exactly.

**Cyclic Quadrilateral:**

A quadrilateral ABCD is called cyclic if all the four vertices of it lie on a circle.

**Common Chord:**

The intersection point of two circles is the common chord of the circle.

### Segment and Sector

– A circular **segment** is a region of a circle which is “**cut off**” from the rest of the circle by a secant or a chord. – **Smaller region **cut off by a chord is called **minor segment** and the **bigger region** is called **major segment**. –

-A **sector** is the portion of a circle **enclosed by two radii and an arc**, where the **smaller area** is known as the **minor sector** and the **larger** being the **major sector**.

– For** 2 equa**l arcs or for semicircles – both the segment and sector is called the** semicircular region.**

### Tangent and Secant

A **line** that **touches** the circle at **exactly one point** is called it’s **tangent**. A **line** that **cuts** a circle at **two points** is called a **secant.**

## Circles and Their Chords

### Theorem of equal chords subtending angles at the centre.

**Proof**: AB and CD are the 2 equal chords.

In Δ AOB and Δ COD

OB = OC [Radii]

OA = OD [Radii]

AB = CD [Given]

ΔAOB ≅ ΔCOD (SSS rule)

Hence, ∠AOB = ∠COD [CPCT]

### Theorem of equal angles subtended by different chords.

– If the** angles** subtended by the chords of a circle at the centre are **equal**, then the **chords are equal.**

Proof: In ΔAOB and ΔCOD

OB = OC [Radii] ∠AOB=∠COD [Given]

OA = OD [Radii]

ΔAOB ≅ ΔCOD (SAS rule)

Hence, AB=CD [CPCT]

### Perpendicular from the centre to a chord bisects the chord.

**Perpendicular **from the **centre** of a circle to a** chord bisects the chord**.

Proof: PQ is a chord and ST is the perpendicular drawn from the centre.

From ΔORP and ΔORQ,

∠ORP=∠ORQ =90^{0} OP = OQ(radii)

OR = OR(common)

Hence, ΔOCB ≅ ΔOCA (RHS rule)

Therefore AC = CB [CPCT]

### A Line through the centre that bisects the chord is perpendicular to the chord.

– A **line drawn** through the centre of a circle to **bisect** a chord is **perpendicular** to the chord.

**Proof: **OC drawn from the center to bisect chord AB.

From ΔOCA and ΔOCB,

OA = OB (Radii)

OC = OC (common)

AC = BC (Given)

Therefore, ΔOMA ≅ ΔOMB (SSS rule)

⇒∠OCA=∠OCB (C.P.C.T)

But, ∠OCA+∠OCB=180^{0}

Hence, ∠OCA=∠OCB=90^{0} ⇒OC⊥AB

### Circle through 3 points

– There is** one **and **only** one **circle** passing through** three given non collinear points.** – A unique circle passes through 3 vertices of a triangle ABC called as the** circum circle. **The **centre** and **radius** are called the **circum center** and **circum radius** of this triangle, respectively.

### Equal chords are at equal distances from the centre.

**Equal chords** of a circle (or of congruent circles) are **equidistant from the centre** (or centres).

NCERT Solution TextBook