Solving Distance and Direction Problems

This question involves tracking movement in different directions to find the final distance from the starting point. We can solve this by breaking down the journey into segments and determining the net displacement in the horizontal (East-West) and vertical (North-South) directions.

Step-by-Step Movement Analysis

Let's assume the woman starts at her home, which we can consider the origin (0,0) on a coordinate plane. Let North be the positive y-axis, South the negative y-axis, East the positive x-axis, and West the negative x-axis.

• Initial Position: Home (0, 0)
• First Movement: Walks 10 km towards the west.
  • Direction: West (negative x direction)
  • Distance: 10 km
  • Change in position: (-10, 0)
  • Current Position: (0 + (-10), 0 + 0) = (-10, 0)
• Second Movement: Turns left and walks 25 km. When facing West, a left turn is towards South.
  • Direction: South (negative y direction)
  • Distance: 25 km
  • Change in position: (0, -25)
  • Current Position: (-10 + 0, 0 + (-25)) = (-10, -25)
• Third Movement: Again turns left and walks 10 km. When facing South, a left turn is towards East.
  • Direction: East (positive x direction)
  • Distance: 10 km
  • Change in position: (10, 0)
  • Current Position: (-10 + 10, -25 + 0) = (0, -25)
• Fourth Movement: After that, she again turns left and walks 5 km. When facing East, a left turn is towards North.
  • Direction: North (positive y direction)
  • Distance: 5 km
  • Change in position: (0, 5)
  • Current Position: (0 + 0, -25 + 5) = (0, -20)

Summarizing the Movements

Calculating the Final Distance from Home

Her final position is (0, -20) relative to her home at (0,0). The distance between two points $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$ is given by the distance formula:

Distance $= \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

In this case, $(x_1, y_1) = (0, 0)$ (Home) and $(x_2, y_2) = (0, -20)$ (Final Position).

Distance $= \sqrt{(0 - 0)^2 + (-20 - 0)^2}$

Distance $= \sqrt{(0)^2 + (-20)^2}$

Distance $= \sqrt{0 + 400}$

Distance $= \sqrt{400}$

Distance $= 20$ km

So, the woman is 20 km away from her house.

Final Answer

Based on our calculation, the woman is 20 km from her house.

Revision Table: Direction and Distance

Additional Information: Solving Direction Problems

Direction and distance problems are common in reasoning tests. They require careful tracking of movements and understanding how turns affect direction. Visualizing the path or using a coordinate system can be very helpful. Always consider the starting point as a reference for calculating the final distance. Sometimes, the question asks for the total distance traveled (sum of all path lengths), but here it asks for the distance from the house (displacement).