Q2 of 59 Page 55

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?



(1) Here OM is the radius of the circle and M and N are the points of contact of MR and NR respectively.


⇒ ∠RMO = 90° Using tangent-radius theorem which states that a tangent at any point of a circle is perpendicular to the radius at the point of contact.


In triangle ORM right-angled at M,


Given that OR = 10 cm and OM = 5 cm {Radius of the circle}


OR2 = OM2 + RM2 {Using Pythagoras theorem}


⇒MR2 = 102 -52


⇒MR2 = 100 - 25


⇒ MR = √75


⇒ MR = 5√3 cm


Also, RN = 5√3 cm {∵ Tangents from the same external point are congruent to each other.}


(2)



⇒∠MRO = 30°


(3) Similarly, ∠NRO = 30°


⇒∠MRN = ∠ MRO + ∠NRO = 30° + 30° = 60°


More from this chapter

All 59 →
1

In the adjoining figure the radius of a circle with centre C is 6 cm, line AB is a tangent at A. Answer the following questions.

(1) What is the measure of ∠CAB ? Why?


(2) What is the distance of point C from line AB? Why?


(3) d(A,B) = 6 cm, find d(B,C).


(4) What is the measure of ∠ABC ? Why?



 3

Seg RM and seg RN are tangent segments of a circle with centre O. Prove that seg OR bisects ∠MRN as well as ∠MON.


 4

What is the distance between two parallel tangents of a circle having radius4.5 cm ? Justify your answer.

 1

Two circles having radii 3.5 cm and 4.8 cm touch each other internally. Find the distance between their centres.