SOLUTION: 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

Algebra ->  Finance -> SOLUTION: 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      Log On


   

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.
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


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.

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.

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).

So the perimeter of the rectangle is

2%28r%2Br%2Asqrt%283%29%29+=+r%282%281%2Bsqrt%283%29%29%29

and the circumference of the circle is

2%28pi%29r

The ratio of the circumference of the circle to the perimeter of the rectangle is then

%282%28pi%29r%29%2F%28r%282%281%2Bsqrt%283%29%29%29%29+=+%28pi%29%2F%281%2Bsqrt%283%29%29