1. What is a Chord?
A chord of a circle is a line segment whose endpoints lie on the circle.
If \(A\) and \(B\) are points on a circle with centre \(O\), then \(AB\) is a chord.
Chords help define arcs, segments, and many important angle theorems.
2. Diameter as the Longest Chord
The diameter is a special chord that passes through the centre of the circle. It is also the longest chord possible in a circle.
2.1. Why the Diameter is Longest
A chord becomes longer as it gets closer to the centre. The diameter lies exactly at the centre, making it the maximum-length chord.
\(\text{Diameter} = 2r\)
3. Equal Chords and Their Properties
Equal chords in a circle always have equal arcs and lie at equal distances from the centre.
3.1. Equal Chords → Equal Distances from Centre
If chords \(AB\) and \(CD\) are equal, then their perpendicular distances from the centre are also equal.
\(AB = CD \Rightarrow OM = ON\)
where \(OM\) and \(ON\) are perpendiculars from the centre to the chords.
3.2. Equal Chords → Equal Arcs
If two chords have the same length, the arcs they subtend (minor arcs) are also equal.
4. Perpendicular Bisector Property
The perpendicular from the centre of a circle to a chord always bisects the chord.
4.1. Important Result
If \(OM \perp AB\), then:
\(AM = MB\)
where \(M\) is the midpoint of chord \(AB\).
4.2. Converse Result
If a line from the centre bisects a chord, then it is perpendicular to that chord.
Both facts are used heavily in geometry proofs and constructions.
5. Distance of a Chord from the Centre
The distance of a chord from the centre helps determine its length. Chords closer to the centre are longer; chords farther from the centre are shorter.
5.1. Length–Distance Relationship
For a circle of radius \(r\), if \(d\) is the perpendicular distance from the centre to the chord of length \(L\), then:
\(L = 2 \sqrt{r^2 - d^2}\)
5.2. Example
In a circle of radius \(10\text{ cm}\), if a chord is \(6\text{ cm}\) away from the centre, then:
\(L = 2\sqrt{10^2 - 6^2} = 2\sqrt{100 - 36} = 2\sqrt{64} = 16\text{ cm}\)
6. Chords and Arcs
Every chord defines an arc. A long chord creates a large arc and a short chord creates a small arc.
6.1. Minor and Major Arcs
A chord divides the circle into a minor arc and a major arc. The minor arc is the smaller one; the major arc is the larger one.
7. Chords in Real-Life Context
Chords appear in many circular structures such as wheels, round windows, and circular designs.
For example, the spokes of a bicycle wheel form chords of the circular rim.