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 length formula
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 in the figure below. 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 length formula
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?
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
Chord of a Circle Theorems
If we try to establish a relationship between different chords and the angle subtended by them in 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. Chord length formula
Theorem 1: Equal Chords Equal Angles Theorem
Statement: Chords which are equal in length subtend equal angles at the center of the circle.
From fig. 3, In ∆AOB and ∆POQ
|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||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 ∠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.
From fig. 4, In ∆AOB and ∆POQ
|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.
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.
|1||AP = AB/2, CQ = CD/2||The perpendicular from centre bisects 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)|