(i) To find the radius of the larger circle, consider the right triangle ABC where AB = r and AC = r. By the Pythagorean theorem:

\(BC^2 = AB^2 + AC^2 = r^2 + r^2 = 2r^2\)

\(BC = \sqrt{2r^2} = r\sqrt{2}\)

Thus, the radius of the larger circle is \(r\sqrt{2}\).

(ii) To find the area of the shaded region:

1. Calculate the area of sector BCFD of the larger circle:

\(\text{Area of sector BCFD} = \frac{1}{4} \pi (r\sqrt{2})^2 = \frac{1}{2} \pi r^2\)

2. Calculate the area of triangle BCAD:

\(\text{Area of } \triangle BCAD = \frac{1}{2} (2r)r = r^2\)

3. Calculate the area of segment CFDA:

\(\text{Area of segment CFDA} = \frac{1}{2} \pi r^2 - r^2\)

4. Calculate the area of semicircle CADE:

\(\text{Area of semicircle CADE} = \frac{1}{2} \pi r^2\)

5. Calculate the shaded area:

\(\text{Shaded area} = \frac{1}{2} \pi r^2 - \left( \frac{1}{2} \pi r^2 - r^2 \right)\)

\(= r^2\)

Thus, the area of the shaded region is \(r^2\).