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 $, m$)?

$AC$ and $BD$ are chords of a circle which bisect each other. Prove that $AC$ and $BD$ are diameters.

Two circles of radii $5\, cm$ and $3\, cm$ intersect at two points and the distance between their centres is $4\, cm.$ Find the length of the common chord. (in $, cm$)

Fill in the blanks:
$(i)$ The centre of a circle lies in . . . . . . of the circle. (exterior/ interior)
$(ii)$ $A$ point,whose distance from the centre of a circle is greater than its radius,lies in . . . . . . of the circle. (exterior/ interior)
$(iii)$ The longest chord of a circle is a . . . . . . of the circle.
$(iv)$ An arc is a . . . . . . when its ends are the ends of a diameter.
$(v)$ Segment of a circle is the region between an arc and . . . . . . of the circle.
$(vi)$ $A$ circle divides the plane,on which it lies,in . . . . . . parts.

Prove that the quadrilateral formed (if possible) by the internal angle bisectors of any quadrilateral is cyclic.

Prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection.