1. **Problem statement:** A cruise ship sails north for 50 km, then turns east and sails another 50 km. We need to find the exact direct distance from the start to the finish of the cruise. 2. **Formula used:** We can model the ship's path as a right triangle where the legs are the distances sailed north and east. The direct distance is the hypotenuse $c$ of the right triangle with legs $a=50$ km and $b=50$ km. 3. **Pythagorean theorem:** $$c = \sqrt{a^2 + b^2}$$ 4. **Substitute values:** $$c = \sqrt{50^2 + 50^2} = \sqrt{2500 + 2500} = \sqrt{5000}$$ 5. **Simplify the square root:** $$\sqrt{5000} = \sqrt{100 \times 50} = 10 \sqrt{50} = 10 \sqrt{25 \times 2} = 10 \times 5 \sqrt{2} = 50 \sqrt{2}$$ 6. **Final answer:** The exact direct distance from start to finish is $$50 \sqrt{2}$$ km.