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 ΔAPR,
AP = RP {Radius of the circle with centre P}
∠PAR = ∠PRA … (1)
In ΔRQB,
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
