Question 1150434
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Draw the figure as described, showing diagonals PR and QS intersecting at O.  Add the two line segments through O parallel to the sides of the rectangle.<br>
The rectangle is now divided into 8 congruent triangles.  With the given measure of angle POQ, all of those triangles are 30-60-90 right triangles.<br>
If the radius of the circle is r, then the lengths of the legs of each of those triangles are r/2 and (r*srqt(3))/2.  That means the lengths of the sides of the rectangle are r and r*sqrt(3).<br>
So the perimeter of the rectangle is<br>
{{{2(r+r*sqrt(3)) = r(2(1+sqrt(3)))}}}<br>
and the circumference of the circle is<br>
{{{2(pi)r}}}<br>
The ratio of the circumference of the circle to the perimeter of the rectangle is then<br>
{{{(2(pi)r)/(r(2(1+sqrt(3)))) = (pi)/(1+sqrt(3))}}}<br>