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

1. What is the length of each tangent segment?
2. What is the measure of ∠MRO?
3. 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°