Question

Find the length of the chord of a circle of radius $6cm$ subtending an angle of ${165}^{\circ }$ at the centre

Answer 1

Let $AB$ be the chord of a circle of radius $6cm$ with $O$ as centre. Draw $OC\perp AB$. Therefore $C$ is the mid point of $AB$ and $\angle AOB={165}^{\circ }$. Then
$\angle AOC=\frac{{165}^{\circ }}{2}={82}^{\circ }{30}^{\prime }$
In the right triangle $OCA$,
$\sin {82}^{\circ }{30}^{\prime }=\frac{AC}{OA}\Rightarrow AC=\sin {82}^{\circ }{30}^{\prime }\times OA$
$AC=0.9914\times 6=5.9484cm$
$\therefore$ Length of the chord $AB=AC\times 2=5.9484\times 2=11.8968cm$