(1)Given RM and RN are the tangents to the circle.

OMR = ONR = 90˚ [Tangent theorem]

In OMR , OR = 10 cm , [Given]

OM = 5cm [Radius]

OR^{2} = OM^{2}+MR^{2} [Pythagoras theorem]

10^{2} = 5^{2}+MR^{2}

MR^{2} = 10^{2}-5^{5}

MR^{2} = 75

MR = 5√3

RN = 5√3 [Tangent segments drawn from an external point to a circle are congruent]

(2)In OMR , OMR = 90˚ [Tangent theorem]

OM = 5cm [Radius]

OR = 10 cm [Given]

OM = ½ OR

MRO = 30˚ …(i) [Converse of 30˚- 60˚ -90˚ theorem]

NRO = 30˚……..(ii)

(3)MRN = MRO +NRO

MRN = 30+30 = 60˚ [from (i) and (ii)]