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