document.write( "Question 1150434: PQRS is a rectangle inscribed in a circle. The circle has centre O and radius r. The angle POQ is 120 degrees. Find the ratio of the circumference of the circle to the perimeter of the rectangle. \n" ); document.write( "

Algebra.Com's Answer #771806 by greenestamps(13203) $\"\"$ $\"About$

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\n" ); document.write( "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.

\n" ); document.write( "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.

\n" ); document.write( "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).

\n" ); document.write( "So the perimeter of the rectangle is

\n" ); document.write( " $\"2%28r%2Br%2Asqrt%283%29%29+=+r%282%281%2Bsqrt%283%29%29%29\"$

\n" ); document.write( "and the circumference of the circle is

\n" ); document.write( " $\"2%28pi%29r\"$

\n" ); document.write( "The ratio of the circumference of the circle to the perimeter of the rectangle is then

\n" ); document.write( " $\"%282%28pi%29r%29%2F%28r%282%281%2Bsqrt%283%29%29%29%29+=+%28pi%29%2F%281%2Bsqrt%283%29%29\"$

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