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

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

(1) It is given that seg RM and seg RN are tangent segments touching the circle at M and N, respectively. 

∴ ∠OMR = ∠ONR = 90º   ...(Tangent at any point of a circle is perpendicular to the radius throught the point of contact)

OM = 5 cm and OR = 10 cm

In right ∆OMR,

OR² = OM² + MR²

MR = `sqrt("OR"^2 - "OM"^2`

MR = `sqrt(10^2 - 5^2)`

MR = `sqrt(100 - 25)`

MR = `sqrt75`

MR = 5`sqrt3` cm

Tangent segments drawn from an external point to a circle are congruent.

∴ MR = NR = 5`sqrt3`

(2) In right ∆OMR,

∠MRO = `"OM"/"MR"`

∠MRO = `(5 "cm")/(5sqrt3cm)`

= `1/sqrt3`

∠MRO = tan 30°

∠MRO = 30°

Thus, the measure of ∠MRO is 30º.

Similarly, ∠NRO = 30º

(3) ∠MRN = ∠MRO + ∠NRO

= 30º + 30º = 60º

Thus, the measure of ∠MRN is 60º.