4 Small Circles

Geometry Level 2

ABCD is a square.

Circle T is inscribed in the square and has a radius R.

The small circles are tangent to the circle T and have radius r.

The fraction r/R can be written as a+b√c.

How much is a+b+c ?


The answer is 3.

Marius Munteanu by Marius Munteanu , 45, United Kingdom

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1 solution

Marius Munteanu
Mar 29, 2014

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5 Helpful 0 Interesting 0 Brilliant 0 Confused

but how does the answer right?

Gary Ibasco - 7 years, 2 months ago

the length of side LV =2R-2r,like that LS=2R-2r also,and VS=2R+2r,which makes LVUS a square.Now LV and LS are its sides and VS is its diagonal,and we know that sqrt 2 side of the square=diagonal or sqrt 2 (2R-2r)=2R+2r or sqrt 2 2(R-r)=2(R+r) or sqrt 2 R+sqrt 2 r=R+r or sqrt 2 R-R=sqrt 2 r+r or (sqrt 2-1) R=(sqrt 2+1) r or (sqrt 2-1)^2 R=(sqrt 2-1) (sqrt 2+1) r or (multiplying by (sqrt 2-1)) or 3-2 sqrt R=r so r/R=3-2 sqrt 2 that makes a+b sqrt c=3-2 sqrt 2,so then a+b+c=3+(-2)+2=3(ans)

Rifath Rahman - 6 years, 11 months ago

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