Starting from point Z, Khanna walked 80 m towards South. He took a left turn and walked 40 m. Then, he took a right turn and walked 40 m. Then, he took a left turn and walked 20 m, Again, he took a left turn and walked 120 m. How far and in which direction is Khanna now from the starting point Z?
60 m, East
Solving Directional Distance Problems: Khanna's Walk
This problem involves tracking a person's movement through a series of turns and calculating their final distance and direction relative to their starting point. We need to follow Khanna's steps carefully, noting each movement and turn.
Understanding the Movement Sequence
Let's break down Khanna's journey step-by-step, starting from point Z:
- Starts at point Z.
- Walks 80 m towards South.
- Takes a left turn and walks 40 m. (When facing South, a left turn is towards the East).
- Takes a right turn and walks 40 m. (When facing East, a right turn is towards the South).
- Takes a left turn and walks 20 m. (When facing South, a left turn is towards the East).
- Takes a left turn and walks 120 m. (When facing East, a left turn is towards the North).
Tracking Position Using Coordinates
To find the final position relative to the starting point Z, let's use a coordinate system. Assume Z is at the origin (0, 0). Let the East direction be the positive x-axis and the North direction be the positive y-axis. Consequently, West is the negative x-axis and South is the negative y-axis.
| Step | Movement | Direction | Distance (m) | Change in (x, y) | Current Position (x, y) |
|---|---|---|---|---|---|
| Start | At Z | - | - | (0, 0) | (0, 0) |
| 1 | Walk | South | 80 | (0, -80) | (0, -80) |
| 2 | Left turn, Walk | East | 40 | (+40, 0) | (0+40, -80+0) = (40, -80) |
| 3 | Right turn, Walk | South | 40 | (0, -40) | (40+0, -80-40) = (40, -120) |
| 4 | Left turn, Walk | East | 20 | (+20, 0) | (40+20, -120+0) = (60, -120) |
| 5 | Left turn, Walk | North | 120 | (0, +120) | (60+0, -120+120) = (60, 0) |
After completing all the steps, Khanna's final position is at coordinates (60, 0). The starting point Z was at (0, 0).
Calculating Final Distance and Direction from Z
The distance from the starting point Z (0, 0) to the final position (60, 0) can be calculated using the distance formula $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
Distance = $\sqrt{(60 - 0)^2 + (0 - 0)^2}$
Distance = $\sqrt{(60)^2 + (0)^2}$
Distance = $\sqrt{3600 + 0}$
Distance = $\sqrt{3600}$
Distance = 60 m
To find the direction, we compare the final position (60, 0) with the starting position (0, 0). The x-coordinate changed from 0 to 60 (a positive change), and the y-coordinate remained 0 (no change). A positive change in the x-coordinate corresponds to the East direction.
Therefore, Khanna is 60 m to the East of the starting point Z.
Summary of Khanna's Path
Khanna's net displacement is 60 m in the East direction from point Z.
Revision Table: Key Concepts
| Concept | Explanation | Relevance to Problem |
|---|---|---|
| Cardinal Directions | North, South, East, West | Essential for understanding movement directions. |
| Turns (Left/Right) | Relative direction change based on current facing direction. | Crucial for determining the direction of subsequent movements. |
| Starting Point | The reference point (Z in this case). | Used to calculate the final distance and direction relative to the origin. |
| Coordinate System | Mapping positions on a 2D plane (x, y). | Helpful method for tracking movements and calculating final position. |
| Distance Formula | $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ | Used to calculate the straight-line distance between two points. |
Additional Information: Solving Direction and Distance Problems
Direction and distance problems are common in reasoning sections of many exams. Here are some tips for solving them effectively:
- Always start by fixing a point (usually the starting point) as a reference.
- Draw a diagram or use a coordinate system to visualize the movement. A diagram can be very helpful for understanding the path and turns.
- Pay close attention to the direction of turns (left or right) as they depend on the current facing direction.
- Break down complex paths into simpler straight-line segments.
- For calculating the final distance and direction from the start, find the net change in position along the East-West axis and the North-South axis.
- Remember that 'distance from the starting point' usually means the shortest straight-line distance (displacement), not the total distance walked.
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Directions: Read the information given below carefully and answer the questions accordingly:
Rashmi stepped out of her college and walked 15 m towards South. She then took a left turn and walked 35 m to reach a pastry shop. She then took a right turn and walked 15 m. She then took a right turn and walked 35 m to reach her house. What is the approximate shortest distance between Rashmi's college and her house via the pastry shop?
Important Questions from Direction and Distance Turns
Starting from her home, a woman walks 10 km towards the west. She turns left and walks 25 km. Again she turns left and walks 10 km. After that, she again turns left and walks 5 km. How far is she from her house now?
A. 35
B. 20
C. 25
D. 40
Lalit walks 9 km east, turns left and walks another 8 km. He again takes a left and walks another 3 km. How far and in which direction is he now from his starting point?
A boy starts from his home northward in order to go to a hotel. He took right turn and took left to reach the hotel. Which direction is the hotel facing?
Arun went 6 m from his home towards south and turned left to cover 8 m. Again he turned left and covered 6 m. He again turned right and covered 4 m. What was the direct distance from his home and which direction was he facing?
Sri walked 4 km towards east from home to school, and he turned three times right side of 6 km respectively. What is the distance between his home and the school?
