Given three circles which share the same tangent and touch each other as shown in the figure, If the radius of the larger Orange Circle is 9 units and that of the smaller Green circle is 4 units. What is the radius of the white circle which lies in between both the circles and also shares the same tangent ?
+128475 We can form a 5-12-13 right triangle by connecting the centers of the circles and then drawing a parallel line to the base of the figure intersecting the two vertical lines in the figure....the length of this line = 12
And we can create two more right triangles
One on the left with legs of (9-r) , x and a hypotenuse of (9 + r)
And one ont the right with legs of (4 - r) (12-x) and (4 + r)
And we have this system of equations
(4 - r)^2 + (12-x)^2 = (4 + r)^2
(9 - r)^2 + (x)^2 = (9 + r)^2
These are a little sticky to solve but using WolframAlpha we find the solutions for x and r to be
x = 36/5 and r = 36/25 = radius of the smaller circle
So....the equation of the small circle is ( x - 36/5)^2 + ( y -36/25) = (36/25)^2
Here's a pic : https://www.desmos.com/calculator/5yzqb20sxi
