Calculating Shyam's Return Path to Point A

This problem requires us to find the shortest distance and direction from Shyam's final location (Point P) back to his starting point (Point A). We can track his journey using coordinates, assuming Point A is the origin (0,0) on a standard map grid, where East is the positive x-axis and North is the positive y-axis.

Step-by-Step Movement Analysis

Shyam's journey can be broken down as follows:

1. Starts at A: Coordinates (0, 0).
2. 12 km East: Moves to (0 + 12, 0) = (12, 0). Facing East.
3. Right turn (South), 7 km: Turns 90 degrees clockwise from East, now facing South. Moves to (12, 0 - 7) = (12, -7). Facing South.
4. Right turn (West), 5 km: Turns 90 degrees clockwise from South, now facing West. Moves to (12 - 5, -7) = (7, -7). Facing West.
5. Left turn (North), 9 km: Turns 90 degrees counter-clockwise from West, now facing North. Moves to (7, -7 + 9) = (7, 2). Facing North.
6. Right turn (East), 7 km: Turns 90 degrees clockwise from North, now facing East. Moves to (7 + 7, 2) = (14, 2). This is the final position, Point P.

So, Shyam's final position P is at coordinates (14, 2) relative to his starting point A (0,0).

Calculating Net Displacement from A to P

The net displacement from A to P combines movements along the East-West (x-axis) and North-South (y-axis) directions:

• East-West displacement: Shyam moved 12 km East, then 5 km West, then 7 km East. The net displacement is $ 12 - 5 + 7 = 14 $ km East.
• North-South displacement: Shyam moved 7 km South (represented as -7) and then 9 km North. The net displacement is $ -7 + 9 = 2 $ km North.

Therefore, Point P is located 14 km East and 2 km North of Point A.

Calculating the Return Journey from P to A

To return to Point A (0, 0) from Point P (14, 2), Shyam must travel the displacement vector PA. This vector is calculated as the coordinates of A minus the coordinates of P:

Vector PA = A - P = $ (0 - 14, 0 - 2) = (-14, -2) $

This vector signifies that Shyam needs to travel 14 km towards the West (negative x-direction) and 2 km towards the South (negative y-direction) from Point P to get back to Point A.

Calculating the Shortest Distance

The shortest distance between two points is the magnitude (length) of the displacement vector connecting them. We use the distance formula, derived from the Pythagorean theorem:

Distance $ = \sqrt{(\Delta x)^2 + (\Delta y)^2} $

Here, $ \Delta x = -14 $ km and $ \Delta y = -2 $ km.

Distance $ = \sqrt{(-14)^2 + (-2)^2} $

Distance $ = \sqrt{196 + 4} $

Distance $ = \sqrt{200} $ km

To simplify the radical:

Distance $ = \sqrt{100 \times 2} = \sqrt{100} \times \sqrt{2} = 10\sqrt{2} $ km

Calculating the approximate numerical value: $ 10\sqrt{2} \approx 10 \times 1.414 = 14.14 $ km.

Determining the Direction

The displacement vector for the return journey is PA = \( (-14, -2) \). The negative component in the x-direction indicates West, and the negative component in the y-direction indicates South. Therefore, the direction from P back to A is South-West.

Summary of Findings

Based on the detailed step-by-step analysis of Shyam's movements, the shortest distance required to return to Point A from Point P is \( 10\sqrt{2} \) km (approximately 14.14 km), and the direction is South-West.

Comparing this calculated result with the options provided:

• 1. 19 km towards east
• 2. 16 km towards north
• 3. 19 km towards north
• 4. 16 km towards south

It appears that none of the provided options accurately match the calculated shortest distance and direction derived from the problem description.