Raju invited friend George for a dinner at his house. When George asked for the direction of
Raju's house, Raju gave him the following instruction .
Proceed 140 metres south from your house then walk 200 metres to east. Then turn to north and walk 100 metres. After that , walk 160 metres to west
What is the shortest distance between the two houses and the direction to Raju's house from George's house ?

Options

A. `40 sqrt(2)` metres and north-west
B. `40 sqrt(2)` metres and south - east
C. 80 metres and south - east
D. 80 metres and north - west

Answer

To find the shortest distance between George's house and Raju's house, we can follow the instructions given by Raju step by step and then apply the Pythagorean theorem.

### Step-by-Step Solution:

1. **Starting Point**: Assume George's house is at the origin (0, 0) on a coordinate plane.

2. **First Instruction**:
- Proceed 140 meters south.
- From (0, 0), moving south means decreasing the y-coordinate.
- New position: (0, -140).

3. **Second Instruction**:
- Walk 200 meters to the east.
- Moving east means increasing the x-coordinate.
- New position: (200, -140).

4. **Third Instruction**:
- Turn to the north and walk 100 meters.
- Moving north means increasing the y-coordinate.
- New position: (200, -40) [since -140 + 100 = -40].

5. **Fourth Instruction**:
- Walk 160 meters to the west.
- Moving west means decreasing the x-coordinate.
- New position: (40, -40) [since 200 - 160 = 40].

6. **Finding the Shortest Distance**:
- Now we have George's house at (0, 0) and Raju's house at (40, -40).
- To find the shortest distance (d) between the two points, we can use the Pythagorean theorem:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
- Here, (x1, y1) = (0, 0) and (x2, y2) = (40, -40).
- So,
\[ d = \sqrt{(40 - 0)^2 + (-40 - 0)^2} = \sqrt{40^2 + (-40)^2} = \sqrt{1600 + 1600} = \sqrt{3200} = 40\sqrt{2} \]

7. **Direction**:
- To find the direction from George's house to Raju's house, we note that Raju's house is located 40 meters east and 40 meters south of George's house.
- Therefore, the direction is South-East.

### Final Answer:
- The shortest distance between George's house and Raju's house is \( 40\sqrt{2} \) meters.
- The direction to Raju's house from George's house is South-East.