A billiard ball is struck by a cue. It travels 100 cm100\,\text{cm}100cm100, start text, c, m, end text before ricocheting off a rail and traveling another 120 cm120\, \text{cm}120cm120, start text, c, m, end text into a corner pocket. The angle between the path as the ball approaches the rail and the path after it strikes the rail is 45∘45^{\circ}45∘45, degrees.
How far is the corner pocket from where the cue initially struck the ball?
Do not round during your calculations. Round your final answer to the nearest centimeter.
cm\text{cm}cmstart text, c, m, end text