The angle subtended at the centre by a chord of a circle radius 6cm is 120°. Find the length of the chord.

Answer Details

In a circle, the angle subtended by a chord at the centre of the circle is twice the angle subtended by the chord at any point on the circumference. Therefore, if the angle subtended by a chord at the centre is 120°, then the angle subtended by the chord at any point on the circumference is 60°.

Consider the triangle formed by joining the endpoints of the chord to the centre of the circle. We know that the angle at the centre of the circle is 120°, and we know that the radius of the circle is 6 cm.

Now we can use trigonometry to find the length of the chord. Let the length of the chord be 2x. Then, in the triangle we have:

\[\sin 60^\circ = \frac{x}{6}\]

\[\Rightarrow x = 6\sin 60^\circ = 6\cdot\frac{\sqrt{3}}{2} = 3\sqrt{3}\]

Therefore, the length of the chord is 2x, which is equal to:

\[2\cdot3\sqrt{3} = 6\sqrt{3}\]

Hence, the answer is (E) \(6\sqrt{3}\) cm.