Understanding Rakesh's Directional Journey

Rakesh's movement can be broken down into distinct steps. We need to track his position relative to his starting point (home) to find the shortest distance.

• Step 1: Rakesh starts at home and walks 5 km South.
• Step 2: He turns right from the South direction (which means he faces West) and walks 2 km West.
• Step 3: He turns right again from the West direction (which means he faces North) and walks 5 km North.
• Step 4: He turns left from the North direction (which means he faces West) and walks 5 km West.

Calculating Net Displacement

To find the shortest distance, we calculate the net displacement in the North-South and East-West directions. Let's consider South and West as negative directions and North and East as positive directions for simplicity, or track them separately.

North-South Displacement:

• Movement South: $5$ km
• Movement North: $5$ km
• Net North-South Displacement = $5 \text{ km (South)} - 5 \text{ km (North)} = 0 \text{ km}$.

This means Rakesh ends up at the same North-South level as his home.

East-West Displacement:

• Movement West: $2$ km
• Movement West: $5$ km
• Net East-West Displacement = $2 \text{ km (West)} + 5 \text{ km (West)} = 7 \text{ km (West)}$.

So, Rakesh's final position is $7$ km directly West of his home.

Determining the Shortest Distance

The shortest distance between two points is a straight line. Since Rakesh's net North-South displacement is $0$ km, his final position is horizontally aligned with his home. The entire displacement is along the East-West axis.

Therefore, the shortest distance between Rakesh's final position and his home is the net East-West displacement.

Shortest Distance = Net East-West Displacement = $7$ km.