Google Circles 2018!
In the diagram below, 3 circles are externally tangential to one another and internally tangential to the largest, 4 circle. Three of these four circles--except for the smallest one--have their centers on a diameter of the largest circle. The second and third largest circles have radii 14 and 7, respectively.
What is the radius of the smallest circle?
The answer is 6.
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Let the radius of circle O 4 be r , the altitude O 4 P from center O 4 to the diameter of circle O 3 be h and O 1 P = a .
Because the circles are tangential to each other, the centers and the tangential point are colinear. And by Pythagorean theorem , we have:
⎩ ⎪ ⎨ ⎪ ⎧ ( 7 + r ) 2 = a 2 + h 2 ( 1 4 + r ) 2 = ( 2 1 − a ) 2 + h 2 ( 2 1 − r ) 2 = ( 1 4 − a ) 2 + h 2 . . . ( 1 ) . . . ( 2 ) . . . ( 3 )
( 2 ) − ( 1 ) : ( 1 4 + r ) 2 − ( 7 + r ) 2 1 4 7 + 1 4 r r + 3 a = ( 2 1 − a ) 2 − a 2 = 4 4 1 − 4 2 a = 2 1 . . . ( 4 )
( 3 ) − ( 2 ) : ( 2 1 − r ) 2 − ( 1 4 + r ) 2 2 4 5 − 7 0 r 5 r + a = ( 1 4 − a ) 2 − ( 2 1 − a ) 2 = − 2 4 5 + 1 4 a = 3 5 . . . ( 5 )
3 × ( 5 ) − ( 4 ) : 1 4 r ⟹ r = 8 4 = 6