# 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?

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.

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.