Practice Set 3.2

- Two circles having radii 3.5 cm and 4.8 cm touch each other internally. Find the…
- Two circles of radii 5.5 cm and 4.2 cm touch each other externally. Find the distance…
- If radii of two circles are 4 cm and 2.8 cm. Draw figure of these circles touching each…
- In fig 3.27, the circles with centres P and Q touch each other at R. A line passing…
- In fig 3.28 the circles with centres A and B touch each other at E. Line is a common…

###### Practice Set 3.2

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

Answer:

Given: Two circles are touching each other internally.

∵The distance between the centres of the circles touching internally is equal to the difference of their radii.

⇒distance between their centres = 4.8 cm – 3.5 cm = 1.3 cm

Question 2.

Two circles of radii 5.5 cm and 4.2 cm touch each other externally. Find the distance between their centres.

Answer:

Given: Two circles are touching each other externally

We know that if the circles touch each other externally, distance between their centres is equal to the sum of their radii.

⇒distance between their centres = 5.5 cm + 4.2 cm = 9.7 cm

Question 3.

If radii of two circles are 4 cm and 2.8 cm. Draw figure of these circles touching each other –

(i) externally

(ii) internally.

Answer:

(i)

__Steps of construction:__

1. Draw a circle with radius 4cm and centre A.

2. Draw another circle with radius 2.8 cm and centre B such that they touch each other externally.

(ii)

__Steps of construction:__

1. Draw a circle with radius 4cm and centre A.

2. Draw another circle with radius 2.8 cm and centre B such that they touch each other internally.

Question 4.

In fig 3.27, the circles with centres P and Q touch each other at R. A line passing through R meets the circles at A and B respectively. Prove that -

(1) seg AP || seg BQ,

(2) ∆ APR ~ ∆ RQB, and

(3) Find ∠ RQB if ∠ PAR = 35°

Answer:

(1)In Î”APR,

AP = RP {Radius of the circle with centre P}

∠PAR = ∠PRA … (1)

In Î”RQB,

RQ = QB {Radius of the circle with centre Q}

∠QRB = ∠QBR …. (2)

⇒ ∠PRA = ∠QRB {Vertically Opposite Angle} ….(3)

⇒ ∠PAR = ∠QBR {From (1), (2) and (3)}

⇒ Alternate interior angles are equal.

⇒ AP || BQ

Hence, proved.

(2) In ∆ APR and ∆ RQB,

∠PAR = ∠QBR and ∠PRA = ∠QRB {From (1) and (2)}

⇒ ∆ APR ~ ∆ RQB {AA}

Hence, proved.

(4) Given: ∠ PAR = 35°

⇒ ∠QBR = 35° = ∠QRB {Proved previously}

In ∆ RQB,

⇒ ∠ RQB + ∠ QRB + ∠QBR = 180° {Angle sum property of the triangle}

⇒∠ RQB + 35° + 35° = 180°

⇒∠ RQB = 180°- 70° = 110°

Question 5.

In fig 3.28 the circles with centres A and B touch each other at E. Line is a common tangent which touches the circles at C and D respectively. Find the length of seg CD if the radii Fig. 3.28 of the circles are 4 cm, 6 cm.

Answer:

Given that two circles with centre A and B touch each other externally. We know that if the circles touch each other externally, distance between their centres is equal to the sum of their radii.

⇒AB = (4 + 6) cm = 10 cm

In ∆ABC right-angles at A,

BC2 = CA2 + AB2 {Using Pythagoras theorem}

⇒BC2 = 42 + 102

⇒BC2 = 16 + 100

⇒ BC = √116 cm

In ∆DBC,

∠ BDC = 90° because D is the point of contact of tangent CD to circle centred B

BC2 = CD2 + DB2 {Using Pythagoras theorem}

⇒CD2 = 116 - 62

⇒CD2 = 116 - 36

⇒ CD = √80 cm = 4√5