Calculating Shortest Distance Between Start and End Points

The problem describes a scenario involving movement in two perpendicular directions (North and West). The shortest distance between the starting point (Z) and the ending point (A) can be found using the Pythagorean theorem, as the path forms a right-angled triangle.

Applying the Pythagorean Theorem

Let Z be the origin (0,0). The man first walks 6 km towards North. His position becomes (0, 6).

Then, he turns left (West) and walks 2 km. His final position A becomes (-2, 6).

The legs of the right-angled triangle are the distance travelled North (6 km) and the distance travelled West (2 km).

The shortest distance 'd' between Z and A is the hypotenuse. According to the Pythagorean theorem:

\(d^2 = (\text{distance North})^2 + (\text{distance West})^2\)

Substitute the given values:

\(d^2 = (6 \text{ km})^2 + (2 \text{ km})^2\)

\(d^2 = 36 \text{ km}^2 + 4 \text{ km}^2\)

\(d^2 = 40 \text{ km}^2\)

Now, find the square root:

\(d = \sqrt{40 \text{ km}^2}\)

Simplify the radical:

\(d = \sqrt{4 \times 10} \text{ km}\)

\(d = \sqrt{4} \times \sqrt{10} \text{ km}\)

\(d = 2\sqrt{10} \text{ km}\)

Therefore, the shortest distance between point Z and point A is \(2\sqrt{10}\) km.