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In the figure there are 3 semicircles touching each other internally and one circle touching 2 of them externally and third one internally. Radius of complete circle is

In the figure there are 3 semicircles touching each other internally and one circle touching 2 of them externally and third one internally. Radius of complete circle is

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Grade:11

1 Answers

Pavan Kumar
32 Points
5 years ago
Hey, thank you for asking the question
without the diagram the explanation is difficult but i will try to explain as best as possible
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

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