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, Mandip to Reshma. If the distance between Reshma and Salma and between Salma and Mandip is 6m each, what is the distance between Reshma and Mandip?

Given : In figure, points R, S and M are showing the position of reshma, salma and mandeep respectively.
Therefore RS = SM = 6 cm
Construction: Join OR, OS, RS, RM, and OM. Draw OL ⊥RS.
Proof: In ΔORS,
OS = OR and OL ⊥RS [∵ By construction]
Therefore, RL = LS = 3 cm [∵ RS = 6 cm]
In ΔOLS, using Pythagoras theorem, OL² = OS² – SL²
⇒ OL² = 5² – 3² = 25 – 9 = 16
⇒ OL = 4
In ΔORK and ΔOMK,
OR = OM [∵ Radii of circle]
∠ROK = ∠MOK [∵ Equal chords subtend equal angle at the centre]
Ok = OK [∵ Common]
Hence, ∠ORK ∠OMK [∵ SAS congruency rule]
RK = MK [∵ CPCT]
Hence, OK ⊥RM
[∵ The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord]
Now, the area of ΔORS = (1/2)× RS × OL …(1)
And the area of ΔORS = (1/2) × OS × KR …(2)
From the equation (1) and (2),
1/2 × RS × OL = 1/2 × OS × KR
⇒ RS × OL = OS × KR ⇒ 6 × 4 = 5 × KR ⇒ KR = (6×4/5) = 4.8
Hence, RM = 2 × KR = 2 × 4.8 = 9.6 cm