Practice Set 3.1

- In the adjoining figure the radius of a circle with centre C is 6 cm, line AB is a…
- In the adjoining figure, O is the centre of the circle. From point R, seg RM and seg RN…
- Seg RM and seg RN are tangent segments of a circle with centre O. Prove that seg OR…
- What is the distance between two parallel tangents of a circle having radius4.5 cm ?…

###### Practice Set 3.1

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?

Answer:

(1) ere CA is the radius of the circle and A is the point of contact of the tangent AB.

⇒ ∠CAB = 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.

(2) CA is the radius of the circle which is perpendicular to the tangent AB.

So, the perpendicular distance of line AB from C = CA = 6 cm

(3) In triangle ABC right-angled at A,

Given AB = 6 cm and CA = 6 cm

BC2 = AB2 + CA2 {Using Pythagoras theorem}

⇒ BC2 = 62 + 62

⇒ BC2 = 36 + 36

⇒ BC = √72

⇒ BC = 6√2 cm

(4) In triangle ABC right-angled at A,

AB = CA = 6 cm

⇒∠ABC = ∠ACB {Angles opposite to equal sides are equal}

⇒∠ABC + ∠ACB + ∠ BAC = 180° {Angle sum property of the triangle}

⇒ 2∠ABC = 90° {∵ ∠ BAC = 90°}

⇒ ∠ABC = 45°

Question 2.

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) 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°

Question 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.

Answer:

In triangle MOR and triangle NOR,

MR = NR {∵Tangents from same external point are congruent to each other.}

OR = OR {Common}

OM = ON {Radius of the circle}

⇒ ΔMOR ≅ ΔNOR {By SSS}

⇒ ∠ROM = ∠RON

And ∠MRO = ∠NRO {C.P.C.T.}

Hence proved that seg OR bisects ∠MRNas well as ∠MON.

Question 4.

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

Answer:

Let BC and DE be the parallel tangents to a circle centered at A with point of contact O and H respectively. On joining OH, we find OH is the diameter of the circle.∠ BOA = 90° = ∠ DHA {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.}

Distance between BC and DE = OH

∵ OH is perpendicular to BC and DE.

OH = 2 × 4.5 cm = 9 cm