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Question
In the adjoining figure, O is the center of the circle. From point R, seg RM and seg RN are tangent segments touching the circle at M and N. If (OR) = 10 cm and radius of the circle = 5 cm, then
(i) What is the length of each tangent segment?
(ii) What is the measure of ∠MRO?
(iii) What is the measure of ∠MRN?
Solution
seg RM and seg RN are tangents to the circle with center O. ....[Given]
∴ ∠OMR = ∠ONR = 90° ...[Tangent theorem]
(i) In ∆OMR,
∠OMR = 90°
∴ OR2 = OM2 + RM2 ...[Pythagoras theorem]
∴ 102 = 52 + RM2
∴ 100 = 25 + RM2
∴ RM2 = 75
∴ RM = `sqrt(75)` ...[Taking square root of both sides]
= `5sqrt(3)` cm
∴ RM = RN ......[Tangent segment theorem]
∴ Length of each tangent segment is `5sqrt(3)` cm.
(ii) In ∆RMO,
∠OMR = 90° ...[Tangent theorem]
OM = 5 cm and OR = 10 cm
∴ OM = `1/2` OR
∴ ∠MRO = 30° .....(i) [Converse of 30°–60°–90° theorem]
Similarly, ∠NRO = 30°
(iii) But, ∠MRN = ∠MRO + ∠NRO ...[Angle addition property]
= 30° + 30° ...[From (i)]
∴ ∠MRN = 60°.
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