Question

In the figure, 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 (where, p and q are natural numbers) then p + q is

Answer 1

Answer:

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 R2R3R and draw a line from RtoR1 it acts as a cevian on this triangle lengths of triangle

R2R3=3

R2R=2+R

R3R=1+R

R1R=3−R [bigger circle is normal and complete circle are same so radius of both coincide]

R2R1=1

R3R1=2

now apply cosine rule at point R1

2×1×(3−r)

(2+r)2−12−(3−r)

2

=

2×2×(3−r)]

−[(1+r)2−22−(3−r)

2

solving this we get r=

14

12

=

7

6

p+q=13