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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, and 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?

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Given: From figure, R, S and, M are the positions 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 $\triangle$ OLS, by Pythagoras theorem,

$OL^2=OS^2-SL^2$

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

$\Rightarrow OL=4$

In $\triangle$ ORK and $\triangle$ OMK,

OR = OM (Radii)

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

OK = OK (Common)

$\triangle$ ORK $\cong$ $\triangle$ OMK (By SAS)

RK = MK (CPCT)

Thus, $OK\perp RM$

area of $\triangle$ ORS = $\frac{1}{2}\times RS\times OL$ ...............................1

area of $\triangle$ ORS = $\frac{1}{2}\times OS\times KR$ .............................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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