Raja's Journey: Calculating the Shortest Path Back to the Starting Point

Imagine a journey taken by Raja, where he drives North for 25 kilometers, turns right, and travels 2 kilometers, then turns right again and drives another 25 kilometers. This route might seem straightforward, but the question arises: how far does Raja have to travel to get back to the starting point?

Let's delve into the geometric calculations behind this journey to determine the shortest distance Raja needs to travel to return home.

Understanding the Journey

Raja's journey consists of three distinct segments:

He starts by driving North for 25 kilometers. Then, he turns right and travels 2 kilometers. Next, he turns right again and drives 25 kilometers.

Based on these movements, Raja's route forms a right-angled triangle, with the first two segments forming the legs of the triangle, and the last segment being the hypotenuse if there is a direct path.

Calculating the Shortest Path

There are two possible routes Raja can take to return to the starting point:

Diagonal Route: If there is a direct diagonal road that connects Raja's final position to the starting point, the distance can be calculated using the Pythagorean theorem. Non-Diagonal Route: If no such diagonal road exists, Raja must retrace his steps partially or take a different route to reach the starting point.

Route 1: Using a Diagonal Road

If there is a diagonal road that connects Raja's final position to the starting point, the distance can be calculated as follows:

First, let's determine Raja's final position after his journey. He travels 25 kilometers North, then 2 kilometers East, and finally, 25 kilometers South. This effectively results in a net northward travel of 35 kilometers and an eastward travel of 2 kilometers. However, the final 25 kilometers South brings him back to the same latitude as his starting point. Therefore, Raja is 10 kilometers north and 2 kilometers east of his starting point. The distance back to the starting point can be calculated using the Pythagorean theorem: Hypotenuse √(10^2 2^2) √104 10.198 KM

Thus, if a diagonal road exists, Raja needs to travel approximately 10.198 kilometers to return to the starting point.

Route 2: Non-Diagonal Route

If no direct diagonal road is available, Raja must follow a different route to return to the starting point. The two possible routes are:

Turning right towards the West and traveling 2 kilometers to rejoin the original route, then traveling south for 10 kilometers to reach the starting point. Alternatively, he could turn left from his final position towards the South and travel 10 kilometers straight.

Both routes result in a total travel distance of 12 kilometers.

Conclusion

Depending on the availability of a direct diagonal road, Raja can choose between two routes to return to the starting point. If a diagonal road exists, the shortest possible distance is approximately 10.198 kilometers. If no such road is available, Raja will need to travel a total of 12 kilometers to reach his starting point.

Understanding the geometric calculations behind these journeys can help in optimizing travel routes in various scenarios. Whether it's in planning routes for delivery services, planning hikes, or simply navigating a city, knowing how to calculate the shortest distance is a valuable skill.