Chord Of A Circle, Its Length and Theorems

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 Circle

Chord Of Circle

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

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 Center Chord Length = 2 × √(r2 − d2)
Chord Lenth Using Trigonometry Chord Length = 2 × r × sin(c/2)
Chord Length of a Circle 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

Example Question Using Chord Length Formula

Question: Find 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, r = 7 cm

and distance, d = 4 cm

Chord length = 2√(r2−d2)

⇒ Chord length = 2√(72−42)

⇒ Chord length = 2√(49−16)

⇒ Chord length = 2√33

⇒ Chord length = 2×5.744

Or , chord length = 11.48 cm

Video Related to Chords

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.

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 ∆AOB and ∆POQ

S.No. Statement Reason
1. AB=PQ Chords of equal length (Given)
2. OA = OB = OP = OQ Radius of the same circle
3. △AOB = △POQ SSS axiom of Congruence
4. ∠AOB = ∠POQ 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∠AOB =∠POQ, then AB=PQ. 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.

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 ∆AOB and ∆POQ

S.No. Statement Reason
1. ∠AOB = ∠POQ Equal angle subtended at centre O (Given)
2. OA = OB = OP = OQ Radii of the same circle
3. △AOB ≅ △POQ SAS axiom of Congruence
4. AB = PQ 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 of a circle are equidistant from the center of the circle.

Equal Chords Equidistant from Center Theorem

S.No. Statement Reason
1 AP = AB/2, CQ = CD/2 The perpendicular from centre bisect the chord
In △OAP and △OCQ
2 ∠1 = ∠2 = 90° OP⊥AB and OQ⊥CD
3 OA = OC Radii of the same circle
4 OP = OQ Given
5 △OPB ≅ △OQD R.H.S. Axiom of Congruency
6 AP = CQ Corresponding parts of congruent triangle
7 AB = CD From statement (1) and (6)

More Topics Related to Chord and Chord Length of Circles

Frequently Asked Questions

What is a Circle?

A circle is defined as a closed two-dimensional figure whose 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 = 2 × √(r2 − d2)

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.

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