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Zorluk: OrtaDirection and Distance Test

A tourist is exploring a historical monument. He starts from the main gate and walks 30 m30\text{ m} towards the West. He then turns 135∘135^{\circ} to his left and walks 202 m20\sqrt{2}\text{ m}. Next, he turns 45∘45^{\circ} to his right and walks 10 m10\text{ m}. Finally, he turns 90∘90^{\circ} to his left and walks 50 m50\text{ m}. What is his shortest distance and direction from the main gate?

  1. 50 m50\text{ m}, South-EastCevap
  2. B
    50 m50\text{ m}, North-East
  3. C
    50 m50\text{ m}, North-West
  4. D
    70 m70\text{ m}, South-East

Cevap

The shortest distance is 50 m50\text{ m} and the direction is South-East.
By tracking the movements sequentially on a 2D coordinate plane from an origin of (0, 0), the tourist's positions are (-30, 0), (-10, -20), (-10, -30), and finally (40, -30). The straight-line distance from the origin to (40, -30) is the hypotenuse of a right triangle with legs of 40 and 30, which equals 50. The coordinate (40, -30) lies in the lower-right quadrant, indicating a South-East direction.

Adım Adım Çözüm

1
Map the starting point as the origin (0, 0) and execute the first movement.
Walking 30 m30\text{ m} West places the tourist at (-30, 0) facing West.
Establishing a Cartesian coordinate system allows for precise tracking of movements.
2
Calculate the second movement: turn 135∘135^{\circ} left and walk 202 m20\sqrt{2}\text{ m}.
A 135∘135^{\circ} left turn from West changes the facing direction to South-East. Moving 202 m20\sqrt{2}\text{ m} South-East adds +20+20 to the X-axis and −20-20 to the Y-axis. New position: (-30 + 20, 0 - 20) = (-10, -20).
A left turn is counter-clockwise. West minus 135∘135^{\circ} equals South-East. Breaking this diagonal vector into components yields equivalent +20 m+20\text{ m} East and −20 m-20\text{ m} South.
3
Calculate the third movement: turn 45∘45^{\circ} right and walk 10 m10\text{ m}.
A 45∘45^{\circ} right (clockwise) turn from South-East points South. Walking 10 m10\text{ m} South subtracts 10 from the Y-axis. New position: (-10, -20 - 10) = (-10, -30).
Right turns add clockwise rotation, shifting the direction from diagonal to a cardinal line.
4
Calculate the final movement: turn 90∘90^{\circ} left and walk 50 m50\text{ m}.
A 90∘90^{\circ} left (counter-clockwise) turn from South points East. Walking 50 m50\text{ m} East adds 50 to the X-axis. Final position: (-10 + 50, -30) = (40, -30).
Standard orthogonal rotation maps the final displacement vector.
5
Calculate the direct distance and direction from the origin (0,0) to the final point (40, -30).
Distance = 402+(−30)2=1600+900=50 m\sqrt{40^2 + (-30)^2} = \sqrt{1600 + 900} = 50\text{ m}. Since X is positive (East) and Y is negative (South), the direction is South-East.
The Pythagorean theorem determines the shortest straight-line Euclidean distance, and the quadrant dictates the cardinal direction.

Anahtar Kavram

Vector displacement combining angular turns, Cartesian mapping, and the Pythagorean theorem.
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Konular

Direction and Distance Test

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