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Anthony starts from Point A and drives 9 km towards West. He takes a left turn, drives 3 km, turns left again and drives 13 km. He then takes a right turn and drives 2 km. He takes a final right turn, drives 4 km and stops at Point P. How far (shortest distance) and towards which direction should he drive in order to reach Point A again? (All turns are 90 degrees turns only unless specified.)
5 km towards the North
13 km towards the West
9 km towards the North
7 km towards the South
5 km towards the North
Track net movements along the Cartesian X and Y axes step by step, then calculate the shortest distance and final direction relative to Point A.
Track net movements along the Cartesian X and Y axes step by step, then calculate the shortest distance and final direction relative to Point A.
Step 1: Trace Movement along West and Turns
Start at Point A (0,0). Drives 9 km West to (-9, 0). Takes a left turn (facing South) and drives 3 km to (-9, -3). Turns left again (facing East) and drives 13 km to (-9 + 13 = 4, -3).
Step 2: Complete the Final Turns to Point P
From (4, -3), takes a right turn (facing South) and drives 2 km to (4, -5). Takes a final right turn (facing West) and drives 4 km to (4 - 4 = 0, -5). Wait, let's re-verify coordinates: Start A(0,0) тЖТ West 9 km тЖТ (-9,0) тЖТ Left (South)тВГ km тЖТ (-9,-3) тЖТ Left (East)тВБтВГ km тЖТ (-9+13 = 4,-3) тЖТ Right (South)тВВ km тЖТ (4,-5) тЖТ Right (West)тВД km тЖТ (4-4=0, -5). Point P is at (0, -5). Wait, let's re-read carefully: Start at A, drive 9 km West (-9,0). Left turn (South)тВГ km тЖТ (-9,-3). Turns left again (East)тВБтВГ km тЖТ (-9+13 = 4,-3). Right turn (South)тВВ km тЖТ (4,-5). Final right turn (West)тВД km тЖТ (4-4=0, -5). Total net coordinates relative to A: X = -9 + 13 - 4 = 0? Wait! Let's check Option A: 5 km towards the North. Let's trace carefully: Start A(0,0) тЖТ West 9 km тЖТ X = -9. Left turn (South) тЖТ Y = -3. Left turn (East)тВБтВГ km тЖТ X = -9 + 13 = +4. Right turn (South)тВВ km тЖТ Y = -3 - 2 = -5. Final right turn (West)тВД km тЖТ X = 4 - 4 = 0. So P is at (0, -5) relative to A? Wait, to reach A(0,0) from (0,-5), he needs to drive 5 km towards the North! Exactly matches Option A!
Step 3: Calculate Distance and Direction
Point P is located 5 km South of Point A. Therefore, to reach Point A from Point P, he must drive 5 km towards the North.
B: 13 km towards the West is incorrect as net horizontal displacement is 0. C: 9 km towards the North is incorrect as the vertical distance is 5 km. D: 7 km towards the South is incorrect in both distance and direction.
A is correct because Anthony stops at Point P which is 5 km South of Point A, hence to reach Point A he must drive 5 km towards the North.
In direction-distance puzzles, always establish a Cartesian grid with a clear origin (0,0) for the starting point to avoid sign errors during multiple left and right turns.
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