One evening, just before sunset, a security guard begins his patrol in an open plaza. He s… — State PSC Exam

One evening, just before sunset, a security guard begins his patrol in an open plaza. He starts walking straight such that his shadow falls exactly to his right. He walks $20\text{ m}$ straight, then turns $135^\circ$ clockwise and walks $15\sqrt{2}\text{ m}$ . Finally, he turns $135^\circ$ counter-clockwise and walks another $15\text{ m}$ straight.

What is the shortest distance between his final position and his starting point, and in which direction is his final position relative to the starting point?

A
$5\text{ m}$ , North
B
$35\text{ m}$ , North-East
$25\text{ m}$ , North-EastAnswer
D
$25\text{ m}$ , South-West

Answer

The shortest distance is

25\text{ m}

and the direction is North-East.

At sunset, the sun is in the West, casting shadows toward the East. Since the guard's shadow is exactly to his right, his right side faces East, meaning he begins by facing North. Walking 20 m North takes him to coordinate (0, 20). He then turns 135 degrees clockwise, which points him South-East. Moving 15√2 m in this direction shifts him 15 m East and 15 m South, placing him at (15, 5). Next, he turns 135 degrees counter-clockwise from South-East, returning him to a North-facing direction. He walks 15 m North, reaching the final coordinate of (15, 20). The shortest distance from the start (0, 0) is found using the Pythagorean theorem: √(15² + 20²) = √(225 + 400) = √625 = 25 m. Because the final position is 15 m East and 20 m North of the origin, the direction is North-East.

Step-by-Step Solution

Determine the initial facing direction based on the sun and shadow.

The guard is facing North.

Since it is evening (sunset), the sun is in the West and casts shadows to the East. If the shadow is to the guard's right, his right side points East, which means he must be facing North.

Calculate the position after the first movement.

Current position is

(0, 20)

Walking

20\text{ m}

straight while facing North changes the Y-coordinate by

+20

Apply the first turn and calculate the second position.

Current position is

(15, 5)

Turning

135^\circ

clockwise from North means the guard now faces South-East. Walking

15\sqrt{2}\text{ m}

South-East yields a horizontal displacement of

+15\text{ m}

(East) and vertical displacement of

-15\text{ m}

(South). New coordinates:

(0+15, 20-15) = (15, 5)

Apply the final turn and calculate the final position.

Final position is

(15, 20)

Turning

135^\circ

counter-clockwise from South-East means subtracting

135^\circ

, placing the guard back to facing North. Walking

15\text{ m}

straight (North) changes the Y-coordinate by

+15

. New coordinates:

(15, 5+15) = (15, 20)

Calculate the shortest distance and direction from the origin

(0,0)

Distance is

25\text{ m}

, Direction is North-East.

Applying the Pythagorean theorem:

\sqrt{15^2 + 20^2} = \sqrt{225 + 400} = \sqrt{625} = 25\text{ m}

. Since both X and Y coordinates are positive, the direction relative to the start is North-East.

Key Concept

Vector displacement combining angular turns, Cartesian coordinates, and shadow-based directional framing.

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