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Difficulty: Very hardDirection and Distance Test

A field researcher starting at a base station walks 15 km15\text{ km} towards the North. He then turns 135∘135^\circ clockwise and walks 82 km8\sqrt{2}\text{ km}. Next, he turns 90∘90^\circ counter-clockwise and walks 72 km7\sqrt{2}\text{ km}. From this location, he turns West and walks 3 km3\text{ km}, and finally turns South and walks 9 km9\text{ km} to reach an observation post. What is the shortest straight-line distance (in km) between the observation post and the base station?

Answer: 13 km

Answer

The shortest straight-line distance between the observation post and the base station is 13 km13\text{ km}.
Resolving each directional segment into horizontal and vertical components gives a net final coordinate of (12,5)(12, 5) relative to the starting origin (0,0)(0, 0). Applying the Pythagorean theorem 122+52=169=13 km\sqrt{12^2 + 5^2} = \sqrt{169} = 13\text{ km} yields the exact straight-line distance.

Step-by-Step Solution

1
Represent the trajectory using 2D coordinate vectors starting from origin (0, 0).
Initial position = (0, 0). After moving 15 km North, position = (0, 15).
North movement aligns with the positive y-axis.
2
Decompose the 135-degree clockwise turn vector (South-East displacement).
Displacement is (+8, -8), placing the researcher at (8, 7).
A 135-degree clockwise turn from North faces South-East (45 degrees below positive x-axis).
3
Decompose the 90-degree counter-clockwise turn vector (North-East displacement).
Displacement is (+7, +7), placing the researcher at (15, 14).
A 90-degree counter-clockwise turn from South-East faces North-East (45 degrees above positive x-axis).
4
Apply cardinal direction adjustments for West and South movements.
Moving 3 km West and 9 km South gives final coordinates (12, 5).
West subtracts from x-coordinate, and South subtracts from y-coordinate.
5
Calculate net displacement using Pythagorean theorem.
Distance = sqrt(12^2 + 5^2) = 13 km.
The direct line between (0, 0) and (12, 5) forms a right-angled triangle with sides 12 and 5.

Key Concept

Vector resolution of cardinal and angular displacements using Cartesian coordinates and the Pythagorean theorem
Estimated Time:3m 0s
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Direction and Distance Test

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