Mathematics

The key insight is that the shortest distance from a point to a line is always perpendicular to that line. Since $P$ must stay on the $x$-axis, you're looking for the point where a perpendicular from $(2,5)$ meets the $x$-axis. Because that perpendicular is vertical, it drops straight down, so $P$ shares the same $x$-coordinate as the target point.

The key insight is that the shortest distance from a point to a line is always perpendicular to that line. Since $P$ must stay on the $x$-axis, you're looking for the point where a perpendicular from $(2,5)$ meets the $x$-axis. Because that perpendicular is vertical, it drops straight down, so $P$ shares the same $x$-coordinate as the target point.