1. **Problem Statement:** Find the length of the line segment $MN$ where $M = (5, 3)$ and $N = (1, -2)$. 2. **Formula:** The distance between two points $M(x_1, y_1)$ and $N(x_2, y_2)$ in the Cartesian plane is given by the distance formula: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ 3. **Substitute the coordinates:** $$d = \sqrt{(1 - 5)^2 + (-2 - 3)^2}$$ 4. **Calculate the differences:** $$d = \sqrt{(-4)^2 + (-5)^2}$$ 5. **Square the differences:** $$d = \sqrt{16 + 25}$$ 6. **Add the squares:** $$d = \sqrt{41}$$ 7. **Simplify:** Since $41$ is a prime number, $\sqrt{41}$ cannot be simplified further. 8. **Conclusion:** The length of $MN$ is $\sqrt{41}$ units. Note: The options given do not include $\sqrt{41}$, so the closest correct answer based on calculation is $\sqrt{52}$ if the problem or points were different, but with the given points, the exact length is $\sqrt{41}$.