Question

A ring, 10 cm in diameter, is suspended from a point 12 cm above its centre by 6 equal strings attached to its circumference at equal intervals. The cosine of the angle between consecutive strings is

• A. $\frac{213}{236}$
• B. $\frac{313}{338}$
• C. $\frac{119}{345}$
• D. None of these

Answer 1

The correct option is B

$\frac{313}{338}$

Let L be the centre of the circle ABCDEF whose diameter be 10 cm. M is the point 12 cm above it and AM, BM, CM, DM, EM, and FM, are strings which hang from M and are attached to it at points A,B,C,D, E and F.
Length of string MB
$=\sqrt{(ML{)}^2+(L{B}^2)}=\sqrt{{12}^2+5^2}=13cm.$
When circumference is divided in six equal parts each chord
= Radius
Hence BC=5cm.

$\cos \left(BMC\right)=\frac{M{B}^2+M{C}^2-B{C}^2}{2.MB\times MC}=\frac{(13{)}^2\times (13{)}^2-(5{)}^2}{2\times 13\times 13}=\frac{313}{338}$