Chords and symmetry

A chord joins two points on a circle; a diameter is a chord through its centre and is the longest chord. A secant cuts the circle at two points. A tangent touches it at one point. Identify the centre from the given information, not from a visual guess.

The perpendicular bisector of a chord passes through the centre. Equivalently, the perpendicular from the centre to a chord bisects it. For a non-diameter chord, the line joining its midpoint to the centre is perpendicular to it. Distances from the centre to chords are always perpendicular distances.

Equal chords in the same circle are equidistant from the centre; conversely, chords at equal perpendicular distances are equal. The same statements hold in circles of equal radius.

Example

A chord is 16 cm long in a circle of radius 10 cm. Let \(M\) be its midpoint. Then half the chord is 8 cm, and its distance from the centre is \(OM=\sqrt{10^2-8^2}=6\text{ cm}\). Every other chord at distance 6 cm from this centre is also 16 cm long. The sketch illustrates the construction, not these dimensions.