Question 197702
If a square is inscribed in a circle with center O and a radius of 2, what is the perimeter of the square?
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Let {{{s}}} repreent a side of the square:

{{{drawing(400,400,-4,4,-4,4, circle(0,0,sqrt(2)), rectangle(-1,-1,1,1), line(-1,-1,0,0), locate(-.7,-.2,2), locate(0,-1,s) )}}}

Draw another radius:

{{{drawing(400,400,-4,4,-4,4, circle(0,0,sqrt(2)), rectangle(-1,-1,1,1), line(-1,-1,0,0), locate(-.7,-.2,2), line(0,0,1,-1), locate(.5,-.2,2), locate(0,-1,s)  )}}}

That is a right triangle. So we use the Pythagorean
theorem to find s:

{{{c^2=a^2+b^2}}}
{{{s^2=2^2+2^2}}}
{{{s^2=4+4}}}
{{{s^2=8}}}
{{{s=sqrt(8)}}}
{{{s=sqrt(4*2)}}}
{{{s=sqrt(4)sqrt(2)}}}
{{{s=2sqrt(2)}}}

So that's the length of a side of the square.
Since there are 4 sides, we multiply by 4 and
the perimeter is

{{{4*2sqrt(2)}}}
{{{8sqrt(2)}}}

Edwin</pre>