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?
- 13 km, South West
- 17 km, South West
- 17 km, North West
- 13 km, West
The Correct Option is B
Approach Solution - 1
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.
Approach Solution -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)
- Forms a right triangle with:
- 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.
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