Question

In the adjoining figure, O is the centre 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 centre 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]

∴ RM = RN [Tangent segment theorem]

Length of each tangent segment is 5 \(\sqrt { 3 }\) cm.

ii. In ∆RMO,

∠OMR = 90° [Tangent theorem]

OM = 5 cm and OR = 10 cm

∴ OM = \(\frac { 1 }{ 2 } \) 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°