Q: 5    Three girls Reshma, Salma and Mandip are playing a game by standing on a circle of radius   \small 5m 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 \small 6m each, what is the distance between Reshma and Mandip?

Answers (1)

Given : From figure,R,S,M are the position of Reshma ,Salma, Mandip respectively.

        So, RS = SM = 6 cm

Construction : Join OR,OS,RS,RM and OM.Draw OL\perp RS.

Proof: 

             

In \triangle ORS,

OS = OR    and  OL\perp RS (by construction )

So,       RL = LS = 3cm        (RS = 6 cm )

In \triangleOLS, by pytagoras theorem,

OL^2=OS^2-SL^2

\Rightarrow OL^2=5^2-3^2=25-9=16

\Rightarrow OL=4

In   \triangle ORK and \triangleOMK,

   OR  = OM  (Radii)

   \angleROK = \angle MOK    (Equal chords subtend equal angle at centre)

         OK = OK         (Common)

     \triangle ORK \cong \triangleOMK    (By SAS)

       RK = MK    (CPCT)

Thus,OK\perp RM

area of \triangleORS  =\frac{1}{2}\times RS\times OL...............................1

area of \triangleORS  =\frac{1}{2}\times OS\times KR.............................2

From 1 and 2, we get

\frac{1}{2}\times RS\times OL  =\frac{1}{2}\times OS\times KR

\Rightarrow RS\times OL=OS\times KR

\Rightarrow 6\times 4=5\times KR

\Rightarrow KR=4.8 cm

Thus, RM =2 KR=2\times 4.8 cm=9.6 cm

 

 

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