In a circle with centre $O$,two chords $PQ$ and $RS$ subtend equal angles at the centre. If $PQ = 12 \text{ cm}$,then $RS = \dots \text{ cm}$.

If two equal chords of a circle intersect,prove that the parts of one chord are separately equal to the parts of the other chord.

Write True or False and justify your answer.
Two chords of a circle of lengths $10 \, cm$ and $8 \, cm$ are at the distances $8.0 \, cm$ and $3.5 \, cm$,respectively,from the centre.

In a circle with centre $O$,the lengths of two chords $AB$ and $CD$ are $12 \, cm$ and $16 \, cm$ respectively. If the chord $AB$ is at a distance $8 \, cm$ from the centre,what is the distance of the chord $CD$ from the centre (in $, cm$)?

The radius of a circle with centre $P$ is $50 \, cm$. Centre $P$ is not lying between two parallel chords $AB$ and $CD$. If $AB = 80 \, cm$ and the distance between $AB$ and $CD$ is $10 \, cm$,then find the length of $CD$. (Given $AB > CD$) (in $, cm$)

Three girls Reshma,Salma,and Mandip are playing a game by standing on a circle of radius $5 \, m$ drawn in a park. Reshma throws a ball to Salma,Salma to Mandip,and Mandip to Reshma. If the distance between Reshma and Salma and between Salma and Mandip is $6 \, m$ each,what is the distance between Reshma and Mandip (in $, m$)?