(a) To find angle XYC, we use the sine rule in the right triangle XYC:

\(\sin(\angle XYC) = \frac{9}{11}\)

Thus,

\(\angle XYC = \sin^{-1}\left(\frac{9}{11}\right) \approx 0.9582 \text{ radians}\)

(b) To find the perimeter of ABC, we calculate each segment:

1. Calculate XY using the Pythagorean theorem in triangle XYC:

\(XY = \sqrt{11^2 - 9^2} = \sqrt{40}\)

2. Calculate AB:

\(AB = 9 + 11 - XY = 20 - \sqrt{40}\)

3. Calculate arc AC:

\(\text{Arc } AC = 11 \times 0.9582 = 10.5402\)

4. Calculate arc BC:

\(\text{Arc } BC = 9 \times \frac{\pi}{2} \approx 14.1372\)

5. Add all segments to find the perimeter:

\(\text{Perimeter} = AB + \text{Arc } AC + \text{Arc } BC = (20 - \sqrt{40}) + 10.5402 + 14.1372 \approx 38.4\)