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

Hint:

Remembering common Pythagorean triplets can help solve distance problems without calculations.
Some standard triplets are:
- \( (3, 4, 5) \)
- \( (5, 12, 13) \)
- \( (8, 15, 17) \)
Here, the legs are 8 and 15, so the hypotenuse is instantly identified as 17.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question involves tracking the movement of a boy starting from his house and determining his final position relative to the starting point.
We need to calculate both the straight-line distance and the direction from the starting point to the ending point.

Step 2: Key Formula or Approach:
We can represent the directions on a standard Cartesian coordinate system where:
- North is along the positive y-axis.
- South is along the negative y-axis.
- East is along the positive x-axis.
- West is along the negative x-axis.
Let the starting point (the house) be the origin \( (0, 0) \).
The total net displacement in the horizontal (East-West) direction is \( \Delta x \), and the total net displacement in the vertical (North-South) direction is \( \Delta y \).
The shortest direct distance \( D \) is calculated using the Pythagorean theorem:
\[ D = \sqrt{(\Delta x)^2 + (\Delta y)^2} \]

Step 3: Detailed Explanation:
1. Let the boy's house be at the origin \( O(0, 0) \).
2. First, the boy travels \( 6\text{ km} \) towards the South.
His position becomes \( (0, -6) \).
3. Second, he travels \( 8\text{ km} \) towards the West.
His position becomes \( (-8, -6) \).
4. Third, he travels \( 9\text{ km} \) further towards the South.
His position becomes \( (-8, -6 - 9) = (-8, -15) \).
5. Now, we analyze the net displacements:
- Total vertical displacement towards South: \( 6\text{ km} + 9\text{ km} = 15\text{ km} \).
- Total horizontal displacement towards West: \( 8\text{ km} \).
6. These displacements form a right-angled triangle where the legs are \( 8\text{ km} \) and \( 15\text{ km} \).
7. We apply the Pythagorean theorem to find the hypotenuse, which represents the direct distance:
\[ D = \sqrt{8^2 + 15^2} \]
8. Computing the squares:
\[ 8^2 = 64 \]
\[ 15^2 = 225 \]
\[ D = \sqrt{64 + 225} = \sqrt{289} \]
9. Since \( 17 \times 17 = 289 \), the direct distance is:
\[ D = 17\text{ km} \]
10. To find the direction, we observe that the boy has moved South and West from the starting point.
Therefore, his direction from the house is South-West.

Step 4: Final Answer:
The boy is at a distance of 17 km in the South-West direction from his house, which corresponds to option (B).