Chord of a Circle, Its Length and Theorems: Chord of a Circle Definition

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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.

Chord of a Circle Definition

Chord of a Circle Definition

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.

Chord of a Circle Definition

Chord of a Circle Definition

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

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 Formula

Chord Length Formula

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

Equal Chords Equal Angles

Equal Chords Equal Angles Theorem

Proof:

From fig. 3, In and

Equal Chords Equal Angles Theorem

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

Equal Angles Equal Chords

Equal Angles Equal Chords Theorem

Proof:

From fig. 4, In and

Equal Angles Equal Chords Theorem

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

Equal Chords Equidistant

Equal Chords Equidistant from Center Theorem

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