alex90 wrote:

The Quadrilateral shown above is a square. Four circles are tangent to the sides of the square and the small circle in the centre is tangent to each of the four circles. What is the ratio of the small circle to the side of the square?

a 1/2

b 1/4(\sqrt{2} -1)

c 1/2(\sqrt{2}-1)

d \sqrt{2} -1

e 2 (\sqrt{2}-2)

Lets assume that the radius of the big circle is 1.

Create a square connecting all the centers (look picture). Its side is 2, it diagonal is \(2\sqrt{2}\), this if formed by 2 radius and the diameter of the smaller circle

\(2\sqrt{2}-2(=2*radiusBig)=diameterSmall\) \(2(\sqrt{2}-1)=diamSmall\)

so radius small = \(\sqrt{2}-1\)

The ratio radius to side square (= 4) is \((\sqrt{2}-1)/4\)

B
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