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?

Answer 1

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°.