Solving Direction and Distance Problems

This problem involves tracking a person's movement through a series of turns and calculating their final distance from a specific point. We will use a coordinate system approach to solve this efficiently. Let's assume Nitin's home is the origin (0,0).

Understanding Nitin's Movement Path

We need to break down Nitin's journey step by step:

• Starts from home (0,0).
• Walks 45 m towards the east.
• Turns left and walks 48 m.
• Turns left again and walks 27 m.
• Takes a final left turn and walks 55 m.

We are asked to find the distance between his final position and a pole located 7 m south of his home.

Step-by-Step Direction Analysis and Position Tracking

Let's track Nitin's position using coordinates, assuming East is along the positive x-axis and North is along the positive y-axis.

1. Starts at Home: Initial Position = (0, 0).
2. 45 m East: Walks 45 m in the positive x direction. Position = (0 + 45, 0) = (45, 0).
3. Turns Left (North) and walks 48 m: From East, a left turn is North (positive y direction). Walks 48 m in the positive y direction. Position = (45, 0 + 48) = (45, 48).
4. Turns Left Again (West) and walks 27 m: From North, a left turn is West (negative x direction). Walks 27 m in the negative x direction. Position = (45 - 27, 48) = (18, 48).
5. Takes a Final Left Turn (South) and walks 55 m: From West, a left turn is South (negative y direction). Walks 55 m in the negative y direction. Position = (18, 48 - 55) = (18, -7).

Nitin's final position is (18, -7) relative to his home at (0,0).

Locating the Pole

The problem states that the pole is exactly 7 m to the south of his home. Since home is at (0,0) and South is the negative y direction, the pole's coordinates are (0, 0 - 7) = (0, -7).

Pole's Position = (0, -7).

Calculating the Distance from Nitin's Final Position to the Pole

Nitin's final position is (18, -7).

The pole's position is (0, -7).

We need to find the distance between these two points. We can use the distance formula, or notice that both points have the same y-coordinate (-7). This means they lie on the same horizontal line (at y = -7). The distance between them is simply the absolute difference in their x-coordinates.

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

Let $(x_1, y_1) = (18, -7)$ and $(x_2, y_2) = (0, -7)$.

Distance = $\sqrt{(0 - 18)^2 + (-7 - (-7))^2}$

Distance = $\sqrt{(-18)^2 + (0)^2}$

Distance = $\sqrt{324 + 0}$

Distance = $\sqrt{324}$

Distance = 18

Alternatively, since the y-coordinates are the same:

Distance = $|x_2 - x_1| = |0 - 18| = |-18| = 18$ m.

The distance between Nitin's final position and the pole is 18 m.

The distance between (18, -7) and (0, -7) is the difference in their x-coordinates, which is 18 - 0 = 18.

So, Nitin is 18 m away from the pole.

Revision Table: Direction and Distance

Understanding directions and turns is crucial for these types of problems:

Additional Information on Coordinate Geometry

Using a coordinate system (like the Cartesian plane) is a powerful method for solving direction and distance problems. We assign directions to axes (e.g., East = +x, North = +y). Each movement can then be represented as a change in the x or y coordinate. The final position is a single point (x, y). The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ can be found using the distance formula derived from the Pythagorean theorem:

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

In cases where the x-coordinates are the same ($x_1 = x_2$), the distance is $|y_2 - y_1|$.

In cases where the y-coordinates are the same ($y_1 = y_2$), the distance is $|x_2 - x_1|$.