A commuter walks 8\text{ km} due South from a train station to reach a park. Facing South,… — State PSC Exam

A commuter walks $8\text{ km}$ due South from a train station to reach a park. Facing South, he then turns left and walks $6\text{ km}$ to reach his office. What is the shortest distance between the train station and his office, and in which direction is the office relative to the train station?

$10\text{ km}$ , South-EastCevap
B
$10\text{ km}$ , South-West
C
$14\text{ km}$ , South-East
D
$10\text{ km}$ , North-East

Cevap

The shortest distance between the train station and the office is

10\text{ km}

, and the office is in the South-East direction relative to the train station.

The initial path

8\text{ km}

South and the subsequent path

6\text{ km}

East form two perpendicular sides of a right triangle. Applying the Pythagorean theorem yields a hypotenuse of

\sqrt{8^2 + 6^2} = 10\text{ km}

. Because the final position is East of the vertical axis and South of the horizontal axis, the direction relative to the start is South-East.

Adım Adım Çözüm

Set up coordinate system and plot the initial movement.

Taking the train station as the origin

(0, 0)

, walking

8\text{ km}

South places the commuter at coordinates

(0, -8)

South corresponds to the negative y-axis direction.

Determine the direction and position after the turn.

Facing South, a left turn reorients the commuter toward the East (+x direction). Walking

6\text{ km}

East moves the position to

(6, -8)

Turning left from South points toward East.

Calculate the shortest distance using the Pythagorean theorem.

\text{Distance} = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10\text{ km}

The straight-line distance from origin

(0,0)

to point

(6,-8)

forms a right-angled triangle with leg lengths 6 and 8.

Determine final direction relative to the starting origin.

Since

x > 0

and

y < 0

, the point

(6, -8)

lies in the South-East quadrant relative to

(0,0)

Positive x (East) and negative y (South) define South-East.

Anahtar Kavram

Pythagorean displacement calculation and relative cardinal direction mapping

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru