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I would like to find the radius of the circumference shown in figure, knowing the side of the square is 5. enter image description here

I have decided to note said radius $r$ and the tiny diagonal bit not included in any circle as $y$.

I think one way to solve this is to find a system composed of two equations and then to solve for $r$ and $y$. However, the only equation I was able to find so far is $$5\sqrt{2}=5+2r+y.$$

How can I find another equation? Or is there a better way to solve this?

enoac
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2 Answers2

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Note that $r+y=\sqrt{2}r$. This can be seen by placing a dot at the centre $C$ of the little circle, and noting that the line segment joining $C$ to the bottom left corner is the diagonal of an $r\times r$ square.

More briefly, $5\sqrt{2}=5+r+\sqrt{2}r$.

André Nicolas
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You already have one equation at hand with you. Now, for the second equation, Assume that you drop a small perpendicular from the centre $C$ of the circle. So that, $r+y$ is the hypotenuse.

So, $\frac{r}{r+y}=\sin 45^\circ$

Or, $\sqrt2r=r+y$

Or, $y=(\sqrt 2-1)r$

Submit that in the first equation to calculate r.

MonK
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