Question

A circular ring of radius $3cm$ is suspended horizontally from a point $4cm$ vertically above the centre by $4$ strings attached at equal intervals to its circumference. If the angle between two consecutive strings be $\theta$, then $\cos \theta$ is equal to

• A. $\frac{4}{5}$
• B. $\frac{4}{25}$
• C. $\frac{16}{25}$
• D. None of these

Answer 1

The correct option is C $\frac{16}{25}$

Let $O$ be the centre of the circular ring which is suspended by the strings $PA,PB,PC$ and $PD$ in such a way that $P$ is just above the point $O$ and arc $AB=$ arc $BC=$ arc $CD=$ arc $AD$
$\therefore \angle AOB={90}^o$
Also, in $△AOP$ we have
$AP=\sqrt{{OA}^2+{OP}^2}=\sqrt{3^2+4^2}=5cm$
$\Rightarrow BA=AP=5cm$
In $△APB$ we have
$\cos \theta =\frac{{AP}^2+{BP}^2-{AB}^2}{2AP.BP}=\frac{5^2+5^2-{\left(3\sqrt{2}\right)}^2}{2\times 5\times 5}$
$\Rightarrow \cos \theta =\frac{16}{25}$