# Chord of a Circle, Its Length and Theorems: Chord of a Circle Definition (For CBSE, ICSE, IAS, NET, NRA 2022)

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## Chord of a Circle Definition

The chord of a circle can be defined as the line segment joining any two points on the circumference of the circle. It should be noted that the diameter is the longest chord of a circle which passes through the center of the circle. The figure below depicts a circle and its chord.

In the given circle with ‘O’ as the center, AB represents the diameter of the circle (longest chord) , ‘OE’ denotes the radius of the circle and CD represents a chord of the circle.

Let us consider the chord CD of the circle and two points P and Q anywhere on the circumference of the circle except the chord as shown. If the endpoints of the chord CD are joined to the point P, then the angle ∠ CPD is known as the angle subtended by the chord CD at point P. The angle ∠ CQD is the angle subtended by chord CD at Q. The angle ∠ COD is the angle subtended by chord CD at the center O.

Angle Subtended by Chord

## Chord Length Formula

There are two basic formulas to find the length of the chord of a circle which are:

Formula to Calculate Length of a Chord | |

Chord Length using Perpendicular Distance from the Centre | Chord Length |

Chord Length using trigonometry | Chord Length |

Chord Length of a Circle Formula

Where,

- r is the radius of the circle
- c is the angle subtended at the center by the chord
- d is the perpendicular distance from the chord to the circle center

### Chord of a Circle Theorems

If we try to establish a relationship between different chords and the angle subtended by them on the center of the circle, we see that the longer chord subtends a greater angle at the center. Similarly, two chords of equal length subtend equal angle at the center. Let us try to prove this statement.

### Theorem 1: Equal Chords Equal Angles Theorem

**Statement**: Chords which are equal in length subtend equal angles at the center of the circle.

Equal Chords Equal Angles Theorem

**Proof**:

From fig. 3, In and

Sr. No. | Statement | Reason |

1. | Chords of equal length (Given) | |

2. | Radius of the same circle | |

3. | SSS axiom of Congruence | |

4. | By CPCT from statement 3 |

**Note**: CPCT stands for congruent parts of congruent triangles.

The converse of theorem 1 also holds true, which states that if two angles subtended by two chords at the center are equal then the chords are of equal length. From fig. 3, if , then . Let us try to prove this statement.

### Theorem 2: Equal Angles Equal Chords Theorem (Converse of Theorem 1)

**Statement**: If the angles subtended by the chords of a circle are equal in measure then the length of the chords is equal.

Equal Angles Equal Chords Theorem

**Proof**:

From fig. 4, In and

Sr. No. | Statement | Reason |

1. | Equal angle subtended at centre O (Given) | |

2. | Radii of the same circle | |

3. | SAS axiom of Congruence | |

4. | From Statement 3 (CPCT) |

### Theorem 3: Equal Chords Equidistant from Center Theorem

**Statement**: Equal chords of a circle are equidistant from the center of the circle.

**Proof**:

**Given**: Chords AB and CD are equal in length.

**Construction**: Join A and C with centre O and drop perpendiculars from O to the chords AB and CD.

Equal Chords Equidistant from Center Theorem

S. No. | Statement | Reason |

1. | , | The perpendicular from centre bisect the chord |

In and | ||

2. | and | |

3. | Radii of the same circle | |

4. | Given | |

5. | R. H. S. Axiom of Congruency | |

6. | Corresponding parts of congruent triangle | |

7. | From statement (1) and (6) |

## Frequently Asked Questions

### What is a Circle?

A circle is defined as a closed two-dimensional figure who՚s all the points in the boundary are equidistant from a single point called its centre.

### What is the Chord of a Circle?

The chord is a line segment that joins two points on the circumference of the circle. A chord only covers the part inside the circle.

### What is the Formula of Chord Length?

The length of any chord can be calculated using the following formula:

Chord Length

### Is Diameter a Chord of a Circle?

Yes, diameter is also considered as a chord of the circle. The diameter is the longest chord possible in a circle and it divides the circle into two equal segments.

### Example Question Using Chord Length Formula

**Question**: Find the length of the chord of a circle where the radius is 7 cm and perpendicular distance from the chord to the center is 4 cm?

**Solution**:

Given radius,

and distance,

Formula for the chord length of a circle,

Chord length

Put the value of diameter and radius.

Chord length

Put the value of square of 7 and 4.

Chord length

Subtract 16 from the 49.

Chord length

Chord length

**Chord length**