The chord theorem

The chord theorem links the length of a chord to the inscribed angle that rests upon it. It is the tool that allows the sine rule to be proved quickly.

Theorem — The chord theorem

In a circle of radius $R$, a chord $AB$ of length $\ell$ is linked to the inscribed angle $\theta$ (subtending the arc not containing $AB$) by the relation: $\ell = 2R\sin\theta.$

This result, already stated in Year 2, finds in Year 4 its most “useful” formulation: it is the basis of the proof of the sine rule and lets us pass smoothly from linear measures (the chords) to angular measures and back.

Links

Topics: Triangle trigonometry
Concepts: Circumscribed circle · Chord theorem
Functions: Sine
Skills: Using formulae