Suraj started running towards West. He walked for 3m, turned left, and ran for 7m. He stopped took a right turn and ran for 300cm. He then again took a right turn and walked for 1200cm. He saw a rabbit towards his right, so he walked in that direction for 9m more. Then he turned right and jumped 2 times each of 50cm. In which direction and how far is he from the starting point?

a) East, 5m

b) North East, 5m

c) North West, 5m

d) North East, 5m

Let's break down Suraj's movement step-by-step and track his position in terms of coordinates:

1. **Starting point: (0,0)**

2. **Moves West 3m**:
   - He moves 3 meters west, so his position is now **(-3, 0)**.

3. **Turns left and runs 7m (North)**:
   - He turns left, which means he is now moving north.
   - He runs 7 meters north, so his position becomes **(-3, 7)**.

4. **Takes a right turn and runs 300cm (East)**:
   - He takes a right turn, so now he is moving east.
   - 300 cm = 3 meters, so his position becomes **(0, 7)**.

5. **Takes another right turn and walks 1200cm (South)**:
   - He takes another right turn, meaning he is now heading south.
   - 1200 cm = 12 meters, so his position becomes **(0, 7 - 12) = (0, -5)**.

6. **Sees a rabbit towards his right and walks 9m (West)**:
   - He turns to his right, which means he is now moving west.
   - He walks 9 meters west, so his position becomes **(-9, -5)**.

7. **Turns right and jumps 2 times each of 50cm (North)**:
   - He turns right again, meaning he is now heading north.
   - 2 jumps of 50 cm = 1 meter, so he moves 1 meter north.
   - His final position becomes **(-9, -4)**.

### Distance and Direction:
- The distance from the starting point (0,0) to the final point (-9, -4) can be calculated using the distance formula:
  \[
  d = \sqrt{(-9 - 0)^2 + (-4 - 0)^2} = \sqrt{81 + 16} = \sqrt{97} \approx 9.85 \text{ meters}
  \]

- The direction is found by calculating the angle between the final position and the origin:
  \[
  \text{Direction} = \text{arctan}\left(\frac{4}{9}\right) \approx 24° \text{ north of west}.
  \]
  Therefore, Suraj is in the **North West** direction.

### Conclusion:
The correct answer is **(c) North West, 5m**. However, the distance calculation in the answer options doesn't match precisely with the actual distance, but based on directions, the closest option is **North West, 5m**.

Would you like more details or have any questions?

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Here are 5 related questions to explore further:
1. How is the distance formula derived?
2. What is the difference between cardinal and intercardinal directions?
3. How do you calculate the angle between two points on a plane?
4. What is the Pythagorean theorem, and how does it relate to distance calculation?
5. How do vector representations help in solving direction-based problems?

**Tip:** Always use a sketch or coordinate system to visualize direction and displacement in problems like this.

Explore how to calculate the direction and distance from the starting point based on Suraj's movement in this geometry problem. Involving concepts like distance formula, coordinate geometry, and trigonometric ratios, this problem is suitable for Grades 6-8 students.

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