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Question

Difficulty: MediumDirection and Distance Test

Four tourists (Alex, Blake, Casey, and Drew) start exploring a flat city from the exact same central plaza. Match each tourist's movement sequence with their final direction relative to the starting point.

  • Alex: Walks 40 m40\text{ m} North, turns 135∘135^\circ clockwise, and walks 402 m40\sqrt{2}\text{ m}.Exactly East
  • Blake: Walks 30 m30\text{ m} West, turns 90∘90^\circ left to walk 30 m30\text{ m}, then turns left again and walks 60 m60\text{ m}.South-East
  • Casey: Walks 20 m20\text{ m} South, turns 90∘90^\circ right and walks 40 m40\text{ m}, then turns 45∘45^\circ clockwise and walks 202 m20\sqrt{2}\text{ m}.Exactly West
  • Drew: Walks 10 m10\text{ m} North, turns 90∘90^\circ right and walks 40 m40\text{ m}, then turns 90∘90^\circ left and walks 30 m30\text{ m}.North-East

Answer

Alex is Exactly East, Blake is South-East, Casey is Exactly West, and Drew is North-East of the starting point.
By placing the starting point at the origin (0,0) of a Cartesian plane, we can translate every movement into (x,y) coordinate changes. Alex ends at (40, 0), indicating a positive x-axis position (East). Blake ends at (30, -30), indicating a positive x and negative y position (South-East). Casey ends at (-60, 0), indicating a negative x-axis position (West). Drew ends at (40, 40), indicating a positive x and positive y position (North-East).

Step-by-Step Solution

1
Map Alex's path on a coordinate plane starting at (0,0).
Moves 40 m40\text{ m} North to (0,40)(0, 40). A 135∘135^\circ clockwise turn from North faces South-East. Moving 402 m40\sqrt{2}\text{ m} SE results in Δx=40\Delta x = 40 and Δy=−40\Delta y = -40. Final position: (40,0)(40, 0).
Vector decomposition (using Pythagoras/trigonometry on 45∘45^\circ right triangles) accurately tracks diagonal displacement.
2
Map Blake's path starting at (0,0).
Moves West 30 m30\text{ m} to (−30,0)(-30, 0). Turning 90∘90^\circ left (faces South) and moving 30 m30\text{ m} gives (−30,−30)(-30, -30). Turning left again (faces East) and moving 60 m60\text{ m} yields (−30+60,−30)=(30,−30)(-30 + 60, -30) = (30, -30).
Step-by-step orthogonal tracking determines the final Cartesian coordinates.
3
Map Casey's path starting at (0,0).
Moves South 20 m20\text{ m} to (0,−20)(0, -20). Turning 90∘90^\circ right (faces West) and moving 40 m40\text{ m} gives (−40,−20)(-40, -20). A 45∘45^\circ clockwise turn from West faces North-West. Moving 202 m20\sqrt{2}\text{ m} NW gives Δx=−20\Delta x = -20, Δy=+20\Delta y = +20. Final position: (−40−20,−20+20)=(−60,0)(-40 - 20, -20 + 20) = (-60, 0).
Translating relative turns (right, clockwise) into cardinal headings ensures correct vector application.
4
Map Drew's path starting at (0,0).
Moves North 10 m10\text{ m} to (0,10)(0, 10). Turning 90∘90^\circ right (faces East) and moving 40 m40\text{ m} gives (40,10)(40, 10). Turning 90∘90^\circ left (faces North) and moving 30 m30\text{ m} yields (40,10+30)=(40,40)(40, 10 + 30) = (40, 40).
Summing the xx and yy vectors gives the exact final location in the first quadrant.

Key Concept

Direction and Distance Vector Mapping
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Topics

Direction and Distance Test

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