Question

A boy leaves his house. He travels 6 km towards South, then travels 8 km towards West and further travels 9 km towards South. How far and in which direction is he from his house now?

• A. 13 km, South West
• B. 17 km, South West
• C. 17 km, North West
• D. 13 km, West

Answer 1

The Correct Option is B

To determine how far and in which direction the boy is from his house, we'll visualize his journey and calculate the resultant displacement.

1. The boy initially moves 6 km south.

2. He then travels 8 km west.

3. Finally, he travels 9 km further south.

This results in a total southward travel of \(6 + 9 = 15\) km.

We now have a right triangle, with the legs measuring 15 km south and 8 km west.

To find the hypotenuse (direct distance from home), we use the Pythagorean theorem:

\[c = \sqrt{a^2 + b^2}\]

where \(a = 15\) km, \(b = 8\) km.

By substituting the values:

\[c = \sqrt{15^2 + 8^2}\]

\[c = \sqrt{225 + 64}\]

\[c = \sqrt{289}\]

\[c = 17\]

Thus, the boy is 17 km away from his house.

For the direction, since he traveled south and west, he is in the south-west direction from his house.

Therefore, the boy is 17 km, South West from his starting point.

Answer 2

The boy is 17 km away from his house in the South-West direction.

Additional Context:

• Movement breakdown:
  • First leg: 6 km South
  • Second leg: 8 km West
  • Third leg: 9 km South
• Net displacement calculation:
  • Southward distance: 6 km + 9 km = 15 km South
  • Westward distance: 8 km West
  • Resultant displacement: √(15² + 8²) = √(225 + 64) = √289 = 17 km
• Direction determination:
  • Both South and West components exist ⇒ South-West direction
  • Angle calculation: tanθ = 8/15 ⇒ θ ≈ 28° West of South
• Visualization:
  • Forms a right triangle with:
    • 15 km as the South leg
    • 8 km as the West leg
    • 17 km as the hypotenuse (actual displacement)
• Common mistakes to avoid:
  • Adding distances linearly (6+8+9=23 km) - incorrect approach
  • Ignoring vector components for direction
  • Confusing North-West with South-West

Correct Answer: (2) 17 km, South West.