Explanation:

To find the ratio of \( R \) to \( r \) in a quarter circle, consider:

- The center of the larger quarter circle is at the origin of the coordinate system.

- The equations require setting up a coordinate system, where:

- The quarter circle has its center at \((R, R)\).

- The inscribed circle's center will be at \((R-r, R-r)\) because it touches the two radii.

- The distance from its center to its touching points is \( r \).

- Applying the Pythagorean theorem in such a setup gives the relationship: \( (R-r)^2 + (R-r)^2 = R^2 \).

- Solving this equation: \( 2(R-r)^2 = R^2 \) leading to \( R = (\sqrt{2} + 1) r \).

Based on these considerations, the correct ratio of \( R \) to \( r \) is:

Option:1 - (\(\sqrt{2} + 1\)) : 1