Question

Question 5
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, 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?

Answer 1

Draw perpendiculars OA and OB on RS and SM respectively.

$AR=AS=\frac{1}{2}\times 6=3m$
OR = OS = OM = 5 m (Radii of the circle)
In $\Delta OAR$,
$O{A}^2+A{R}^2=O{R}^2$[using pythagoras theorem]
$O{A}^2+(3m{)}^2=(5m{)}^2$
$O{A}^2=(25-9)m^2=16m^2$
OA = 4 m

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.

Let the diagonals of the kite ORSM bisects each other at a point C.
$\therefore \angle RCSwillbeof{90}^{\circ }andRC=CM$

$Areaof\Delta ORS=\frac{1}{2}\times OA\times RS=\frac{1}{2}\times 4\times 6=12m^2$ ------> (i)
Also, $Areaof\Delta ORS=\frac{1}{2}\times RC\times OS=\frac{1}{2}\times RC\times 5$-------> (ii)
From (i) and (ii),
$\frac{1}{2}\times RC\times 5=12$
$RC=\frac{12\times 2}{5}=4.8m$
RC =4.8m

We have, RM =RC +CM
We know that RC = CM (given)
$\therefore$ RM =2RC =2(4.8)=9.6m
Therefore, the distance between Reshma and Mandip is 9.6 m.