Solving the Direction and Distance Problem

This question asks us to find the minimum distance between Manish's starting point and his ending point after a series of movements based on directions and turns.

Understanding the Movements

Let's break down Manish's path step by step:

• Initially, Manish is facing South.
• He takes a 90° right turn. If you are facing South, a right turn means turning towards the West. So, he is now facing West.
• He walks 8 km in the West direction. Let's assume his starting point is the origin (0,0) on a coordinate plane. Walking 8 km West takes him to the point (-8, 0).
• From this point, he turns right again. He is currently facing West. A right turn from West means turning towards the North. So, he is now facing North.
• He walks 6 km in the North direction. From the point (-8, 0), walking 6 km North takes him to the point (-8, 6).

Determining Start and End Points

• Starting Point: (0, 0)
• Ending Point: (-8, 6)

The minimum distance between the starting point and the ending point is the straight-line distance, which is the displacement.

Calculating Minimum Distance using Pythagorean Theorem

The path taken by Manish (8 km West followed by 6 km North) forms two sides of a right-angled triangle. The 8 km movement is along the West-East axis, and the 6 km movement is along the North-South axis. These two paths are perpendicular to each other, forming the two shorter sides (legs) of a right-angled triangle. The minimum distance between the start and end points is the hypotenuse of this triangle.

We can use the Pythagorean theorem to find the length of the hypotenuse (\(c\)), given the lengths of the two legs (\(a\) and \(b\)). The theorem states:

\(c^2 = a^2 + b^2\)

In this case:

• Leg a = 8 km (distance moved West)
• Leg b = 6 km (distance moved North)
• Hypotenuse c = Minimum distance between start and end points

Substituting the values into the theorem:

\(c^2 = (8 \, \text{km})^2 + (6 \, \text{km})^2\)

\(c^2 = 64 \, \text{km}^2 + 36 \, \text{km}^2\)

\(c^2 = 100 \, \text{km}^2\)

To find \(c\), we take the square root of both sides:

\(c = \sqrt{100 \, \text{km}^2}\)

\(c = 10 \, \text{km}\)

So, the minimum distance between Manish's starting point and ending point is 10 km.

Summary of the Path

The straight line from (0,0) to (-8,6) is the minimum distance, which is 10 km.

Revision Table: Key Concepts in Direction and Distance

Additional Information: Directions and Geometry

Problems involving direction and distance often require visualizing movements and using basic geometry, particularly the Pythagorean theorem, to find the shortest path or displacement.

• Visualisation: Always try to draw a diagram or mentally trace the path taken. Starting at an origin point (like 0,0) and using coordinates can be very helpful, especially when movements are along the cardinal directions.
• Perpendicular Movements: When movements are perpendicular (like North-South and East-West), they naturally form the legs of a right-angled triangle with the displacement as the hypotenuse. This is where the Pythagorean theorem is crucial for finding the minimum distance.
• Understanding Turns: A standard "right turn" or "left turn" usually implies a 90° turn unless specified otherwise. Knowing which direction results from a turn from a given facing direction is fundamental (e.g., Right from South is West, Right from West is North).
• Distance vs. Displacement: The total distance walked is the sum of the lengths of all segments of the path (8 km + 6 km = 14 km in this case). The minimum distance or displacement is the straight-line distance from the start to the end point (10 km here). The question specifically asks for the minimum distance.