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The problem in the image describes the bus's journey in several steps, involving turns and movements in different directions. Here's a breakdown of the movements to find the shortest distance:

1. **Bus's path:**
   - Starts at point \( P \).
   - Moves east for 2 km to reach point \( Q \).
   - Takes two consecutive left turns:
     - First, 3 km to reach point \( R \).
     - Then, 2 km to reach point \( S \).
   - Takes a right turn of 45 degrees and moves 10 km to reach point \( T \).
   - Takes another right turn of 135 degrees and moves 6 km to reach point \( U \), directly above \( S \).

2. **Goal:**  
   Find the shortest distance from point \( P \) (the starting point) to point \( U \).

Let me calculate the shortest distance from \( P \) to \( U \).The shortest distance from point \( P \) to point \( U \) is 9 km. However, since this value isn't one of the available answer options, it may suggest a small rounding adjustment or the problem's hidden assumptions.

Would you like further clarifications or details?

Here are five related questions to explore:
1. What role do angles play in determining the final position of the bus?
2. How do you calculate distances when multiple turns are involved?
3. How would the solution change if the bus made different angle turns?
4. Why is the shortest distance often a straight-line distance in such problems?
5. How can trigonometry be used to solve movement problems like this one?

**Tip:** In geometry problems, always consider if there’s symmetry or patterns in the angles or distances to simplify the calculation process.

A bus starts its journey from point P. It moves towards the east direction for 2 km and reaches Q, from there it takes two consecutive left turns to reach R and S, which are 3 km and 2 km away respectively. After reaching S, the bus takes a 45 degrees right turn and moves 10 km to reach T, then it takes a 135 degrees right turn and moves 6 km to reach U, which is exactly above S. Find the shortest distance of the bus from its starting point.

This problem explores a bus's movement with various turns and angles, using trigonometry and geometry to calculate the shortest distance from its starting point to the final position. Learn how to apply Pythagoras' theorem and angle-based movements to solve distance problems. Suitable for Grades 9-10.

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