A geometry problem by Shamin Yeaser

1) Use the above result to obtain the relationship between the radius of the smaller circle (r) and the larger circle (R) as R = 4r

2) Next use Pythagoras theorem to obtain the result (R+ r)^2 - R^2 = (90/2)^2 = 45^2

Solve for R and find 2R. ( R = 60 and r = 15)

WELL,this may need some drawings in the picture and figure out some equations! It is given that,the distance between the center of smaller circles are 90.lets,determine their radius as x and the radius of the bigger circles as r
So, 2r=90+2x....................(i)
Connecting the three points.(the tangent point of two ciclres,the center of Bigger circle and the center of Smaller circle? we can have a right angled triangle having r+x as the hypotenuse, r and $\frac{90}{2}$ =45 as other sides.Using pythagorean theorem,
* (r+x)²=r²+45²
or, r²+2rx+x²=r²+2025
or, 2rx+x²=2025 or, x(2r+x)=2025
or, x(90+2x+x)=2025 or, x(90+3x)=2025
or, 3x²+90x-2025=0 *
solving * x , we find x=15 or x=-45 * . So, we have the small circle radius x=15 .substituting x in equation(i),
2r=90+2x=90+2*15=90+30=120 . And guess what? this is what we looking for!! the DISTANCE!! (which is 2r)