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There are 3 semicircles touching each other internally and one circle touching two of them externally and third one internally. Radius of the complete circle (in its lowest form) is p/q p, q belong to N then p + q is ?

There are 3 semicircles touching each other internally and one circle touching two of them externally and third one internally. Radius of the complete circle (in its lowest form) is 
p/q
p, q belong to N
then p + q is ?

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Grade:10

1 Answers

Arun
25750 Points
5 years ago
Dear student
 
 
let center of bigger semicircle(radius=3) is R1 ,the next bigger one (radius=2) is R2 and the smaller one(radius=1) is R3 and complete circle one is R (assume its radius is r )
Now consider the triangle R2 R3 R and draw a line from R to R1 it acts as a cevian on this trisngle lengths of triangle
R2 R3 = 3
R2 R =2+R
R3 R=1+R
R1 R=3-R  [bigger circle is normal and complete circle are same so radius of both coincide]
R2 R1=1
R3 R1=2
now apply cosine rule at point R1
(2+r)2-12-(3-r)2/2*1*(3-r)   = -[(1+r)2-22-(3-r)2/2*2*(3-r)]
solving this we get   r=  12/14=6/7
p+q=13
 
Regards
Arun

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