The key chord property
A chord joins two points on a circle. There is an important symmetry: if you draw a line from the centre that is perpendicular to a chord, that line cuts the chord into two equal halves. In other words, the perpendicular from the centre bisects the chord.
Key idea / Key idea
If OM is perpendicular to chord AB (O is the centre, M on AB), then AM = MB. The reverse is also true: the line from the centre to the midpoint of a chord is perpendicular to it.
Equal chords
Two chords that are the same length are the same distance from the centre. Chords nearer the centre are longer; chords farther from the centre are shorter.
Finding lengths with Pythagoras
Because the perpendicular makes a right angle, we can use the Pythagorean theorem. The radius, half the chord, and the distance from the centre to the chord form a right-angled triangle.
Worked example / Worked example
A chord AB is 16 cm long in a circle of radius 10 cm. Find the distance from the centre O to the chord.
The perpendicular bisects AB, so AM = 16 ÷ 2 = 8 cm.
In right triangle OMA: OA² = OM² + AM².
10² = OM² + 8² → 100 = OM² + 64 → OM² = 36 → OM = 6 cm.
Remember / Remember
- Perpendicular from centre bisects the chord.
- Line from centre to a chord's midpoint is perpendicular to it.
- Equal chords are equidistant from the centre.
- Use radius² = distance² + (half-chord)².