A chord of a circle has a length of 6 cm. The angle subtended by the chord at a point on the circumference is 30°. What is the distance from the center of the circle to the chord?

Solution

Solution: The central angle subtended by the chord is twice the angle at the circumference, so it is \(2\times30^\circ=60^\circ\). The chord length is 6 cm, giving radius \(r=6\) cm because \(6=2r\sin30^\circ\). The perpendicular from the centre bisects the chord, so half the chord is 3 cm. The distance from the centre is \(\sqrt{6^2-3^2}=\sqrt{27}=3\sqrt3\) cm. Trick: Use the central-angle theorem, then use Pythagoras with the radius and half-chord.