The Karch-Randall braneworld provides a framework to study Hawking radiation using holographic methods. A black hole on the brane is coupled to a non-gravitational bath, allowing computation of Hawking radiation entropy via the quantum extremal surface prescription. The Ryu-Takayanagi surface in the ambient space corresponds to the quantum extremal surface, with a topological phase transition indicating the Page curve of Hawking radiation consistent with unitarity. This paper fills a gap by deriving the quantum extremal surface prescription in the Karch-Randall braneworld. For (2+1)-dimensional ambient space, the topological phase transition of the Ryu-Takayanagi surface corresponds to the formation of a replica wormhole on the brane. In higher dimensions, the brane satisfies Einstein's equation with conformal matter. The paper discusses implications for gravitational path integrals and generalizes results to higher dimensions. The Ryu-Takayanagi conjecture is proven in the AdS/BCFT context, showing that the minimal area surface corresponds to the entanglement island. Replica wormholes emerge as dominant contributions in the path integral, leading to the formation of entanglement islands. The paper concludes with discussions on the implications of these results for quantum gravity and holography.The Karch-Randall braneworld provides a framework to study Hawking radiation using holographic methods. A black hole on the brane is coupled to a non-gravitational bath, allowing computation of Hawking radiation entropy via the quantum extremal surface prescription. The Ryu-Takayanagi surface in the ambient space corresponds to the quantum extremal surface, with a topological phase transition indicating the Page curve of Hawking radiation consistent with unitarity. This paper fills a gap by deriving the quantum extremal surface prescription in the Karch-Randall braneworld. For (2+1)-dimensional ambient space, the topological phase transition of the Ryu-Takayanagi surface corresponds to the formation of a replica wormhole on the brane. In higher dimensions, the brane satisfies Einstein's equation with conformal matter. The paper discusses implications for gravitational path integrals and generalizes results to higher dimensions. The Ryu-Takayanagi conjecture is proven in the AdS/BCFT context, showing that the minimal area surface corresponds to the entanglement island. Replica wormholes emerge as dominant contributions in the path integral, leading to the formation of entanglement islands. The paper concludes with discussions on the implications of these results for quantum gravity and holography.