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


Answer :

Let O be the centre of the circle and Reshma, Salma and Mandip are represented by the points R, S and M, respectively. Let RP = x m.

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Area of \(\triangle{ORS}\) = \(\frac{1}{2} \) × x × 5 = \(\frac{5x}{2} \) ...(i)
(\(\because \) in \(\triangle{ORM}\), RM is a chord
\(\therefore \) \(OP\perp{RM}\))

In right angled triangle \(\triangle{RON}\),
\({OR}^2 = {RN}^2 + {NO}^2\)
\(\Rightarrow \) \({5}^2 = {3}^2 + {NO}^2\)
\(\Rightarrow \) \({NO}^2 = 25 - 9 = 16\)
\(\therefore \)NO = 4cm
So, Area of \(\triangle{ORS}\) = \(\frac{1}{2} \) × RS × ON = \(\frac{1}{2} \) × 6 × 4 = 12 ...(ii)

From Equations (i) and (ii), we get,

\(\Rightarrow \) \(\frac{5x}{2} \) = 12
\(\therefore \) x = \( \frac{24}{5} \) .

Since, P is the mid- Point of RM,

Thus, RM = 2RP = 2 × \(\frac{24}{5} \)
\(\Rightarrow \) RM = \(\frac{48}{5} \) = 9.6m

Hence, the distance between Reshma and Mandip is 9.6m.

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