Radius !!

Geometry Level 3

If the radius of each yellow circle is r r , express the radius of large red circle in terms of r r .

r ( r + 3 ) r(r + \sqrt 3) r 2 r\sqrt 2 2 r ( 1 + 2 ) 2r(1+\sqrt 2) r ( 1 + 2 ) r(1+\sqrt 2)

Naren Bhandari by Naren Bhandari , 21, India

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1 solution

Join the centers of the small circles to form a square. Let d d be the diagonal of this square. By pythagorean theorem, d = 2 r 2 d=2r\sqrt{2} . Let D D be the diameter of the big circle. From the figure, D = 2 r + d = 2 r + 2 r 2 D=2r+d=2r+2r\sqrt{2} . So the radius of the big circle is,

R = D 2 = 2 r + 2 r 2 2 = r + r 2 = r ( 1 + 2 ) R=\dfrac{D}{2}=\dfrac{2r+2r\sqrt{2}}{2}=r+r\sqrt{2}=\boxed{r(1+\sqrt{2})}

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