Question

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

Answer 1

Draw perpendiculars $OA$ and $OB$ on $RS$ and $SM$ respectively
Let $R,S$ and $M$ be the position of Reshma, Salma and Mandip respectively.
$AR=AS=\frac{6}{2}=3m$
$OR=OS=OM=5m$ (Radii of the circle)
In $\Delta OAR$,
$O{A}^2+A{R}^2=O{R}^2$
$O{A}^2+(3m{)}^2=(5m{)}^2$
$O{A}^2=(25-9)m^2=16m^2$
$OA=4m$
$ORSM$ will be a kite ($OR=OM$ and $RS=SM$).
We know that the diagonals of a kite are perpendicular and the diagonal common to both the isosceles triangles is bisected by another diagonal.
$\angle RCS$ will be of ${90}^{\circ }$ and $RC=CM$
Area of $\Delta ORS=\frac{1}{2}\times OA\times RS$
$\frac{1}{2}\times RC\times OS=\frac{1}{2}\times 4\times 6$
$RC\times 5=24$
$RC=4.8$
$RM=2RC=2\left(4.8\right)=9.6$
Therefore, the distance between Reshma and Mandip is $9.6m$.