Equal Chords and Their Distances from the Centre

A circular park of radius $20 \mathrm{m}$ is situated in a colony. Three boys Ankur, Syed and David are sitting at equal distance on its boundary each having a toy telephone in their hands to talk to each other. Find the length of the string of each phone.

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

If a line intersects two concentric circles (circles with the same centre) with centre $O$ at $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and $\mathrm{D},$ prove that $\mathrm{A}\mathrm{B}=\mathrm{C}\mathrm{D}.$

If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords.

If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.

Two circles of radii $5 \mathrm{c}\mathrm{m}$ and $3 \mathrm{c}\mathrm{m}$ intersect at two points and the distance between their Centres is $4 \mathrm{c}\mathrm{m}.$ Find the length of the common chord.