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Zorluk: OrtaDirection and Distance Test

A field researcher starts from a central station, initially facing North. He walks 12 m12\text{ m} due North, turns 90∘90^\circ clockwise, and walks 9 m9\text{ m}. He then turns 135∘135^\circ clockwise and walks 152 m15\sqrt{2}\text{ m} straight. From this position, he turns 45∘45^\circ anti-clockwise and walks 5 m5\text{ m}. Finally, he turns 90∘90^\circ anti-clockwise and walks 12 m12\text{ m} to reach a monitoring post. What is the shortest straight-line distance (in meters) between the central station and the monitoring post?

Cevap: 10 m

Cevap

The shortest straight-line distance between the central station and the monitoring post is 10 m10\text{ m}.
By resolving each displacement into xx (East-West) and yy (North-South) components, the final position relative to the starting central station is (6,−8)(6, -8). Applying the Pythagorean theorem 62+(−8)2\sqrt{6^2 + (-8)^2} yields 10 m10\text{ m}.

Adım Adım Çözüm

1
Represent the movements as vector displacements on a Cartesian coordinate plane with origin (0,0)(0, 0) at the central station.
Initial position is (0,0)(0, 0) facing North.
Cartesian grid helps systematically evaluate multi-turn directional paths.
2
Calculate position after leg 1 and leg 2.
Leg 1 (12 m North): position is (0,12)(0, 12), facing North. Leg 2 (90° clockwise = East, 9 m): position is (9,12)(9, 12), facing East.
Moving North increases yy-coordinate; turning 90° clockwise points East, increasing xx-coordinate.
3
Calculate position after leg 3.
Turning 135° clockwise from East sets direction to South-West (225∘225^\circ). Displacement vector is (−152cos⁡45∘,−152sin⁡45∘)=(−15,−15)(-15\sqrt{2} \cos 45^\circ, -15\sqrt{2} \sin 45^\circ) = (-15, -15). New position is (9−15,12−15)=(−6,−3)(9 - 15, 12 - 15) = (-6, -3), facing South-West.
135° clockwise turn from East (90°) points along the South-West diagonal.
4
Calculate position after leg 4.
Turning 45° anti-clockwise from South-West sets direction to South (180∘180^\circ). Walking 5 m South gives new position (−6,−3−5)=(−6,−8)(-6, -3 - 5) = (-6, -8), facing South.
45° anti-clockwise turn from South-West (225°) rotates direction to South (180°).
5
Calculate position after leg 5.
Turning 90° anti-clockwise from South sets direction to East (90∘90^\circ). Walking 12 m East gives final position (−6+12,−8)=(6,−8)(-6 + 12, -8) = (6, -8).
90° anti-clockwise turn from South rotates direction to East, increasing xx-coordinate by 12.
6
Compute the shortest straight-line distance from origin (0,0)(0, 0) to final position (6,−8)(6, -8).
Distance d=62+(−8)2=36+64=100=10 md = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10\text{ m}.
Shortest distance between two points on a plane is given by the Pythagorean distance formula.

Anahtar Kavram

Shortest distance calculation using 2D vector coordinate mapping and angle turns
Tahmini Süre:1m 30s
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Direction and Distance Test

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