In a quarter circle of radius R, a circle of radius r is inscribed. What is the ratio of R to r?

Explanation

An inscribed circle in a quarter circle touches the two straight radii and the arc. Connecting the centers creates a right-angled triangle where the hypotenuse is the distance between the two centers, measuring r\sqrt{2}. Extending this line from the inscribed circle's center to the arc adds r. Thus, the total distance (which is the radius R of the quarter circle) is r\sqrt{2} + r = r(\sqrt{2}+1). The ratio R:r = (\sqrt{2}+1):1.

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