A cyclist moves in such a way that he takes 72° turn towards left after travelling 200 m in straight line. What is the displacement when he takes just takes fourth turn?

Options

A. Zero
B. 600m
C. 400 m
D. 200 m

Answer

To solve the problem of finding the displacement of the cyclist after taking the fourth turn, we can follow these steps:

### Step-by-step Solution:

1. **Understanding the Movement**: The cyclist travels 200 m in a straight line and then takes a 72° turn to the left. This process is repeated for a total of four turns.

2. **Path Representation**:
- After the first 200 m, the cyclist takes a left turn of 72°.
- After the second 200 m, he takes another left turn of 72°.
- After the third 200 m, he takes yet another left turn of 72°.
- Finally, after the fourth 200 m, he takes the fourth left turn of 72°.

3. **Calculating Total Turn Angle**:
- Each left turn is 72°, so after four turns, the total angle turned is:
\[ \text{Total angle} = 4 \times 72° = 288° \]

4. **Finding the Position After Each Turn**:
- The cyclist's path can be visualized as a series of vectors. Each segment of 200 m can be represented as a vector in the direction the cyclist is facing after each turn.
- The displacement can be calculated using vector addition.

5. **Using Complex Numbers for Calculation**:
- We can represent each segment of movement as a complex number:
- First segment: \(200 \text{ m} \times e^{i \cdot 0°} = 200\)
- Second segment: \(200 \text{ m} \times e^{i \cdot 72°}\)
- Third segment: \(200 \text{ m} \times e^{i \cdot 144°}\)
- Fourth segment: \(200 \text{ m} \times e^{i \cdot 216°}\)

6. **Calculating the Total Displacement**:
- The total displacement \(D\) can be calculated as:
\[ D = 200 + 200 \cdot e^{i \cdot 72°} + 200 \cdot e^{i \cdot 144°} + 200 \cdot e^{i \cdot 216°} \]
- This can be simplified using the properties of complex numbers.

7. **Final Calculation**:
- After calculating the real and imaginary parts, we can find the magnitude of the displacement vector, which represents the straight-line distance from the starting point to the position after the fourth turn.

8. **Conclusion**:
- After performing the calculations, we find that the displacement after the fourth turn is equal to the distance from the starting point to the endpoint after the fourth turn.