To determine the length of the line segment that begins at the point \( (5, -2) \) and ends at the point \( (5, 7) \), we can use the distance formula for two points in a Cartesian coordinate system, which is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
In this case, we have:
- Point 1 \( (x_1, y_1) = (5, -2) \)
- Point 2 \( (x_2, y_2) = (5, 7) \)
Now, plug these values into the distance formula:
\[ d = \sqrt{(5 - 5)^2 + (7 - (-2))^2} \] \[ d = \sqrt{(0)^2 + (7 + 2)^2} \] \[ d = \sqrt{0 + (9)^2} \] \[ d = \sqrt{81} \] \[ d = 9 \]
Therefore, the length of the line segment is \( \boxed{9} \).