A person walks 2 km towards East and then he turns to South and walk 1 km. Again he turns to East and walks 2 km after that he turns to North and walk 4 km. Now how far is he from his starting point ?

Question

Accepted Answer

To determine how far the person is from the starting point, we need to trace the path taken and calculate the resultant distance.

1. **Initial Movement:**
   - The person walks 2 km towards the East.

2. **Second Movement:**
   - The person turns to the South and walks 1 km.

3. **Third Movement:**
   - The person turns to the East again and walks 2 km.

4. **Fourth Movement:**
   - The person turns to the North and walks 4 km.

Now, let's analyze the movements step by step:

- **Eastward Movement:** The person initially moves 2 km East and then another 2 km East, totaling 4 km East.
- **Southward Movement:** The person moves 1 km South.
- **Northward Movement:** The person moves 4 km North.

To find the final position relative to the starting point, we calculate the net displacement in the North-South and East-West directions:

- **Net East-West Displacement:** 
  - Total Eastward movement = 4 km
  - There is no Westward movement, so the net East-West displacement is 4 km East.

- **Net North-South Displacement:**
  - Total Northward movement = 4 km
  - Total Southward movement = 1 km
  - Net North-South displacement = 4 km North - 1 km South = 3 km North

Now, we use the Pythagorean theorem to calculate the straight-line distance from the starting point:

$$
\text{Distance} = \sqrt{(\text{East-West displacement})^2 + (\text{North-South displacement})^2}
$$

$$
\text{Distance} = \sqrt{(4 \text{ km})^2 + (3 \text{ km})^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \text{ km}
$$

Therefore, the person is 5 km away from the starting point.

**Explanation of Incorrect Options:**

- **Option 2 (6 km):** This would imply a greater displacement than calculated. The Pythagorean theorem clearly shows the resultant distance is 5 km, not 6 km.
- **Option 3 (3 km):** This would suggest a much shorter displacement, which does not account for the actual path taken and the resultant vector calculated.
- **Option 4 (4 km):** This would imply a direct line of movement without considering the actual path and resultant vector, which is incorrect as per the calculations.

Thus, the correct answer is Option 1, which is 5 km.