Math101 · Independent Ontario learning library
Math101Chords
A chord is a segment whose endpoints lie on a circle. A diameter is the longest chord because it passes through the centre.
Chord theorems connect linear distances, arcs, and angles and support circle constructions, engineering cross-sections, and proofs.
Intuition and core definition
A chord is a segment whose endpoints lie on a circle. A diameter is the longest chord because it passes through the centre. In the same circle, equal chords subtend equal central angles and arcs, and chords equidistant from the centre are congruent.
Notation, language, and conditions
If radius $r$, perpendicular centre-to-chord distance $d$, and chord length $c$ are related, the perpendicular from the centre bisects the chord. A right triangle gives $(c/2)^2+d^2=r^2$, requiring $0\le d\le r$.
Why this idea matters
A radius perpendicular to a chord bisects it, turning chord-length questions into right-triangle calculations.
A dependable method
- Draw the centre and chord and mark known lengths.
- Drop a perpendicular from the centre to the chord.
- Use the theorem that this perpendicular bisects the chord.
- Apply the Pythagorean theorem to one of the congruent right triangles.
- Double a half-chord result and compare with diameter $2r$.
