Study the information given below carefully and answer the questions that follow: Kartick starts moving in a car from point A. He moves 30 km to the south and then turns left from point B and moves 50 km till point C. He then turns right and moves 60 km and then turns left from point D and moves 50 km till point E. He then turns right from point E and moves 20 km and then turns right from point F. He moves 150 km and then turns right from point G and moves 80 km till point H and stopped.
In which direction is point D with respect to point H?
Kartick's Journey: Tracing the Path and Understanding Turns
To accurately determine the direction of point D with respect to point H, we must first trace Kartick's entire journey step-by-step. We will assign coordinates to each significant point, starting from point A, to visualize their exact positions on a two-dimensional plane. We assume positive X is East, negative X is West, positive Y is North, and negative Y is South.
Coordinate Plotting: Locating Each Point Accurately
Let's assume Kartick starts at Point A, which we place at the origin of our coordinate system (0,0). We then track his movements and turns to find the coordinates of each point (B, C, D, E, F, G, H).
| Segment | Movement Description | Direction | Distance | Start Point Coordinates | End Point Coordinates |
|---|---|---|---|---|---|
| 1 | Kartick starts from A, moves 30 km South | South ($\downarrow$) | 30 km | A (0,0) | B (0, -30) |
| 2 | From B, turns left, moves 50 km till C | East ($\rightarrow$) | 50 km | B (0, -30) | C (0 + 50, -30) = (50, -30) |
| 3 | From C, turns right, moves 60 km | South ($\downarrow$) | 60 km | C (50, -30) | D (50, -30 - 60) = (50, -90) |
| 4 | From D, turns left, moves 50 km till E | East ($\rightarrow$) | 50 km | D (50, -90) | E (50 + 50, -90) = (100, -90) |
| 5 | From E, turns right, moves 20 km till F | South ($\downarrow$) | 20 km | E (100, -90) | F (100, -90 - 20) = (100, -110) |
| 6 | From F, turns right, moves 150 km till G | West ($\leftarrow$) | 150 km | F (100, -110) | G (100 - 150, -110) = (-50, -110) |
| 7 | From G, turns right, moves 80 km till H | North ($\uparrow$) | 80 km | G (-50, -110) | H (-50, -110 + 80) = (-50, -30) |
Direction Calculation: Point D Relative to Point H
Now that we have the coordinates for all the points, we can determine the direction of point D with respect to point H. This means we imagine ourselves standing at point H and looking towards point D.
- Coordinates of Point D: $D = (50, -90)$
- Coordinates of Point H: $H = (-50, -30)$
To find the relative position of D from H, we subtract the coordinates of H from the coordinates of D:
Relative X-coordinate of D from H:
$\Delta X = X_D - X_H = 50 - (-50) = 50 + 50 = 100$
Relative Y-coordinate of D from H:
$\Delta Y = Y_D - Y_H = -90 - (-30) = -90 + 30 = -60$
So, the relative position of point D with respect to point H is $(100, -60)$.
- Since the $\Delta X$ (X-coordinate difference) is positive ($+100$), point D is to the East of point H.
- Since the $\Delta Y$ (Y-coordinate difference) is negative ($-60$), point D is to the South of point H.
Resulting Direction: Point D from Point H
When an object is to the East and South of a reference point, its direction is classified as South-east.
Therefore, point D is in the South-east direction with respect to point H.
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