A person at a point D on a straight road AC has four options to go to B which lies on a perpendicular to AC through D. Which one of the following is the shortest route to B ?

For any point B and a straight line AC, the shortest route from B to any point on AC is along the perpendicular from B to AC (the foot of the perpendicular gives the minimum distance). Therefore, when minimizing the total D→X + X→B for X constrained to lie on AC, the minimum occurs at the foot of the perpendicular from B onto AC. Among the four given choices, E is the point on AC that is the foot of this perpendicular, so the path D→E→B is shortest. This follows from standard geometry results on distance from a point to a line and reflection/minimization arguments that show the perpendicular gives the minimum length [1].

Sources

1. [1] https://brilliant.org/wiki/distance-between-point-and-line/