In a morning after sunrise, a boy rode his bicycle 4 km towards west. Then the took right turn and rode 6 km then he right turn and rode 6 km to reach is school. In which direction the school is from then starting point?
North - East
Understanding the Bicycle Journey and Direction
The question asks for the final direction of the school relative to the starting point, given a series of movements made by a boy on his bicycle after sunrise.
Step-by-Step Movement Analysis
Let's break down the boy's journey starting from an initial point. We can represent the starting point as the origin \( (0,0) \) on a coordinate plane, where the positive x-axis is East, the negative x-axis is West, the positive y-axis is North, and the negative y-axis is South.
- Step 1: The boy starts at \( (0,0) \) and rides 4 km towards the west. Moving 4 km west means decreasing the x-coordinate by 4. The new position is \( (0 - 4, 0) = (-4, 0) \). After this movement, the boy is facing West.
- Step 2: He then takes a right turn. If someone is facing West, a right turn means turning towards North. He rides 6 km in this new direction (North). Moving 6 km north means increasing the y-coordinate by 6, while the x-coordinate remains -4. The new position is \( (-4, 0 + 6) = (-4, 6) \). After this movement, the boy is facing North.
- Step 3: He then takes another right turn. If someone is facing North, a right turn means turning towards East. He rides 6 km in this new direction (East). Moving 6 km east means increasing the x-coordinate by 6, while the y-coordinate remains 6. The final position (where the school is) is \( (-4 + 6, 6) = (2, 6) \).
Let's summarize the movements and positions:
| Step | Direction | Distance | Change in Position | Current Position | Facing Direction |
|---|---|---|---|---|---|
| Start | - | - | - | \( (0,0) \) | Implicit |
| 1 | West | 4 km | \( \Delta x = -4, \Delta y = 0 \) | \( (-4, 0) \) | West |
| 2 | North (Right from West) | 6 km | \( \Delta x = 0, \Delta y = +6 \) | \( (-4, 6) \) | North |
| 3 | East (Right from North) | 6 km | \( \Delta x = +6, \Delta y = 0 \) | \( (2, 6) \) | East |
Determining the Final Direction from Starting Point
The starting point was \( (0,0) \) and the school is located at \( (2, 6) \). To find the direction of the school from the starting point, we look at the coordinates of the final position relative to the origin:
- The x-coordinate is \( +2 \). Since the positive x-axis represents East, this means the school is 2 km to the East of the starting point.
- The y-coordinate is \( +6 \). Since the positive y-axis represents North, this means the school is 6 km to the North of the starting point.
A point that is both to the East and North of the starting point lies in the North-East direction from the origin.
Therefore, the school is in the North-East direction from the starting point.
Revision Table: Key Concepts in Direction Problems
| Concept | Explanation | How it applies here |
|---|---|---|
| Cardinal Directions | North, South, East, West are the four main directions. | Used to define movement vectors. |
| Relative Directions | Right/Left turns change the facing direction relative to the current one. | A right turn from West is North; a right turn from North is East. |
| Displacement Vector | The straight-line distance and direction from start to end. Calculated as \( (\Delta x, \Delta y) \). | Final position \( (2,6) \) relative to \( (0,0) \) gives displacement East and North. |
| Coordinate System | Using \( (x,y) \) to track position relative to an origin. | Used effectively to track the boy's position after each step. |
Additional Information: Visualizing the Journey
Imagine drawing this journey on a map or graph paper:
- Start at the center.
- Draw a line 4 units left (West).
- From there, draw a line 6 units up (North).
- From there, draw a line 6 units right (East).
The final point you reach will be to the right and up from your starting point, which corresponds to the North-East direction. Even though the distances travelled were 4 km, then 6 km, then 6 km, the straight-line displacement from the start \( (0,0) \) to the school \( (2,6) \) determines the direction.
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