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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
- What is the length of each tangent segment?
- What is the measure of ∠MRO?
- What is the measure of ∠MRN?
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Solution
(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.
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