PROBLEMS ON INTERNALLY AND EXTERNALLY TANGENT CIRCLES

Problem 1 :

In the picture given below. CD is tangent of both circles.

AB  =  20, AC  =  15 and BD  =  10. Find CD.

Solution :

In triangle ABE,

AC  =  DE  =  15

So, x  =  BE  =  5

<AEB  =  90

Using Pythagorean theorem,

AB²  =  AE²+BE²

20²  =  AE²+5²

400-25  =  AE²

AE  =  √375

AE  =  19.3

AE  =  CD  =  19.3

So, length of common tangent is 19.3 cm.

Problem 2 :

AB  =  24, AC  =  18 and CD  =  19. Find BD.

Solution :

Let BE  =  x.

In triangle ABE,

<AEB  =  90 degree

AB²  =  AE²+EB²

24²  =  19²+x²

576 - 361  =  x²

x  =   √215

x  =  14.66

BD  =  DE-BE

BD  =  18-14.66

BD  =  3.34

So, BD is 3.34.

Problem 3 :

AC is tangent to both circles. Find the measure of angle ∠CQB

If AO=9 and AB=15

Solution :

In triangles OAB, BCQ

<OAB  =   <BCQ  (90 degree)

<OBA  =  <CBQ  (Vertically opposite angles)

So,

<AOB  =  <CQB

In triangle OAB,

OA  =  9 and AB  =  15

OB²  =  OA² + AB²

OB²  =  9² + 15²

OB  =  √(81+225)

OB  =  √306

OB  =  17.49

cos θ  =  Adjacent / Hypotenuse

cos <AOB  =  OA/OB

cos <AOB  =  9/17.49

cos <AOB  =  0.514

<AOB  =  cos^-1(0.514)

<AOB  =  59.06

<AOB  =  59  =  <CQB

So, the required angle is 59.

Problem 4 :

Find x, if DC  =  2x+3 and EC  =  x+10

Solution :

Length of tangents drawn from out of the circle will be equal.

So,

DC  =  EC

2x+3  =  x+10

x  =  7

Problem 5 :

Four identical coins are lined up in a row as shown. The distance between the centers of the first and the fourth coin is 42inches. What is the radius of one of the coins? 

Solution :

Let "r" be the radius of one circle.

r+2r+2r+r  =  42

6r  =  42

r  =  42/6

r  =  7

So, radius of one circle is 7 cm.

Problem 6 :

Four circles are arranged inside an equilateral triangle as shown. If the triangle has sides equal to 16cm , what is the radius of the bigger

Solution :

In triangle ADC

tanθ  =  Opposite side/Adjacent side

tan 60  =  DA/DO

√3  =  8/DO

DO  =  8/√3

DO  =  4.618

Approximately 4.62.

So radius if the larger circle is 4.62 cm.

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