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Difficulty: MediumDirection and Distance Test

An automated delivery drone is tested along a flight path starting from point PP. It first flies 12 m12\text{ m} due North to reach point QQ. At point QQ, it turns 90∘90^\circ to its right and flies 9 m9\text{ m} to reach point RR. From point RR, it rotates 135∘135^\circ clockwise and flies 152 m15\sqrt{2}\text{ m} in a straight line to reach point SS. Finally, it turns due North and flies 3 m3\text{ m} to land at pad TT. What is the shortest distance and direction of landing pad TT relative to the starting point PP?

  1. 6 m6\text{ m} due WestAnswer
  2. B
    6 m6\text{ m} due East
  3. C
    15 m15\text{ m} due South-West
  4. D
    3 m3\text{ m} due West

Answer

The shortest distance of landing pad T relative to point P is 6 meters due West.
Using Cartesian coordinates centered at the starting point, the drone moves to (0, 12) at point Q and then to (9, 12) at point R. Rotating 135 degrees clockwise from facing East reorients the drone towards South-West. Moving 15√2 meters in the South-West direction yields a displacement of -15 meters along the x-axis and -15 meters along the y-axis, bringing point S to (-6, -3). Finally, moving 3 meters North adds 3 to the y-coordinate, ending at point T with coordinates (-6, 0). Relative to the origin (0, 0), point T is exactly 6 meters to the West.

Step-by-Step Solution

1
Establish Cartesian coordinate system with starting point P at origin (0, 0).
P is at (0, 0).
Setting an origin allows precise vector tracking.
2
Calculate position of Q after moving 12 meters North, and R after moving 9 meters East.
Q = (0, 12), R = (9, 12).
North adds to y-coordinate; turning 90 degrees right from North heads East, adding to x-coordinate.
3
Calculate position of S after rotating 135 degrees clockwise from East (which points South-West).
x-displacement = -15sqrt(2) * cos(45°) = -15; y-displacement = -15sqrt(2) * sin(45°) = -15. S = (9 - 15, 12 - 15) = (-6, -3).
A 135-degree clockwise turn from East reorients direction to South-West.
4
Calculate final position T after flying 3 meters North from S(-6, -3).
T = (-6, -3 + 3) = (-6, 0).
Flying North increases the y-coordinate by 3.
5
Determine distance and direction of T(-6, 0) relative to P(0, 0).
Distance = 6 meters; Direction = West.
The x-coordinate is -6 and y-coordinate is 0, which lies 6 units along the negative x-axis (West).

Key Concept

Direction and Distance Test - Angular Rotations and Vector Displacement
Estimated Time:1m 30s
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Topics

Direction and Distance Test

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