Understanding the Man's Journey

Let's trace the path the man takes from his starting point 'O' and determine the coordinates of each point mentioned. We can assume 'O' is at the origin (0,0) of a coordinate system. East is the positive x-direction, West is the negative x-direction, North is the positive y-direction, and South is the negative y-direction.

• Starts at 'O': (0,0)
• Travels 20 km towards East to reach point 'A': From (0,0) move 20 units in the positive x direction. Point 'A' is at (20, 0).
• Turns right (South) and travels 10 km to reach point 'B': From (20,0) move 10 units in the negative y direction. Point 'B' is at (20, -10).
• Turns right (West) and travels 9 km to reach point 'C': From (20,-10) move 9 units in the negative x direction. Point 'C' is at (20 - 9, -10) = (11, -10).
• Turns right (North) and travels 5 km to reach point 'D': From (11,-10) move 5 units in the positive y direction. Point 'D' is at (11, -10 + 5) = (11, -5).
• Turns left (West) and travels 11 km, stops, and travels 1 km to reach point 'E': This and the subsequent movement to 'F' are not needed to find the distance between 'A' and 'D'.
• Turns right (North) and travels 6 km to reach point 'F': Not needed for the A to D distance calculation.

We are asked to find the shortest distance between point 'A' and point 'D'.

Point 'A' has coordinates (20, 0).

Point 'D' has coordinates (11, -5).

Calculating Shortest Distance Between Point A and Point D

The shortest distance between two points in a coordinate system can be found using the distance formula, which is derived from the Pythagorean theorem.

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\( \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)

Here, let's take Point 'A' as \((x_1, y_1) = (20, 0)\) and Point 'D' as \((x_2, y_2) = (11, -5)\).

Substitute these coordinates into the distance formula:

\( \text{Distance AD} = \sqrt{(11 - 20)^2 + (-5 - 0)^2} \)

Calculate the differences in the x and y coordinates:

\( \text{Distance AD} = \sqrt{(-9)^2 + (-5)^2} \)

Square the differences:

\( \text{Distance AD} = \sqrt{81 + 25} \)

Add the squared values:

\( \text{Distance AD} = \sqrt{106} \)

The shortest distance between point 'A' and point 'D' is \(\sqrt{106}\) km.

Revision Table: Journey Points

Additional Information: Distance and Displacement

In this problem, we calculated the "shortest distance" between two points, which is essentially the magnitude of the displacement vector between them. This is different from the total distance traveled.

• Distance: The total length of the path traveled. In this case, the total distance from O to D is 20 km + 10 km + 9 km + 5 km = 44 km.
• Displacement: The straight-line distance between the starting and ending points, along with direction. The shortest distance refers to the magnitude of this displacement.

Understanding coordinate geometry is crucial for solving these types of directional distance problems, as it allows you to represent movements in different directions mathematically and use geometric formulas like the distance formula.