## TRICKY MATHS PUZZLES – PART – I

### Here in this blog we are discussing about TRICKY  MATHS  PUZZLES –  PART – I . It is illustrated below :

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1. The bigger circle radius is 4 cm and the medium circle radius is 2 cm . The line tangent to all the circles and the three circles touch each other . Then what is the radius of the smallest circle ?

SOLUTION :

Let the radius of the bigger ( orange ) circle be 4 cm and the medium ( blue ) circle

be  2 cm

Here r is the radius of the smaller (red ) circle   The distance( X )  between the orange and the red circle will be obtained from having the distances (4 + r) and ( 4 – r ) and the corresponding distance ( Y ) blue and red circles from  ( 2 + r ) and  ( 2 – r  )

By Pythagoras theorem , we have

X =  √ ( 4 + r ) 2 –  (  4 – r ) 2

= √  ( 16 + r2 + 8r ) – ( 16 + r2 – 8r )

=  √  8r + 8r    =  √ 16 r

X =  4√ r

Y  =√ ( 2 + r ) 2 – ( 2 – r ) 2

=  √(  4 + r2 + 4r ) – ( 4 + r2 – 4r )

= √ 4r + 4r   =  √8 r

Y    =  2 √2 r

x + y  = √ r ( 4 + 2√ 2 )

Thus by Pythagoras theorem we get ,

6 2           =  2 2 + { (4 + 2√2 ) √r } 2

36 – 4  =   { ( 16 + 4(2) + 16 2 ) r }

32   =   { 16 + 8 + 16 * 1.414 } r

=   {  24 + 22.62 } r

32  =    46. 62 r

r     =  32 / 46.62

=     0.6864 cm

Hence the radius of smaller circle is r = 0.68 cm

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